%------------------------------------------------------------------------------
% File     : ITP244^1 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Succ 00501_031916
% 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    : 0070_VEBT_Succ_00501_031916 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11150 (5897 unt; 999 typ;   0 def)
%            Number of atoms       : 27368 (12446 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 113930 (2501   ~; 461   |;1721   &;99167   @)
%                                         (   0 <=>;10080  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :   88 (  87 usr)
%            Number of type conns  : 4183 (4183   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  915 ( 912 usr;  61 con; 0-8 aty)
%            Number of variables   : 25521 (2184   ^;22596   !; 741   ?;25521   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 00:38:24.252
%------------------------------------------------------------------------------
% Could-be-implicit typings (87)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    product_prod_nat_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    list_P4002435161011370285od_o_o: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    list_list_nat: $tType ).

thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    list_VEBT_VEBT: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
    product_prod_o_int: $tType ).

thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
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thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
    set_list_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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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__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__Rat__Orat_J,type,
    set_rat: $tType ).

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

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

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

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

thf(ty_n_t__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__Rat__Orat,type,
    rat: $tType ).

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

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

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

% Explicit typings (912)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: rat > int ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Otake__bit__num,type,
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thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
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thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
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thf(sy_c_Code__Numeral_Obit__cut__integer,type,
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thf(sy_c_Code__Numeral_Odivmod__abs,type,
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thf(sy_c_Code__Numeral_Odivmod__integer,type,
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thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
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thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
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thf(sy_c_Code__Numeral_Ointeger__of__num,type,
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thf(sy_c_Code__Numeral_Onat__of__integer,type,
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thf(sy_c_Code__Numeral_Onum__of__integer,type,
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thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
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thf(sy_c_Complex_OArg,type,
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thf(sy_c_Complex_Ocis,type,
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thf(sy_c_Complex_Ocnj,type,
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thf(sy_c_Complex_Ocomplex_OComplex,type,
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thf(sy_c_Complex_Ocomplex_OIm,type,
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thf(sy_c_Complex_Ocomplex_ORe,type,
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thf(sy_c_Complex_Ocsqrt,type,
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thf(sy_c_Complex_Oimaginary__unit,type,
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thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Divides_Oadjust__mod,type,
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thf(sy_c_Divides_Odivmod__nat,type,
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thf(sy_c_Divides_Oeucl__rel__int,type,
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    if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).

thf(sy_c_If_001t__Int__Oint,type,
    if_int: $o > int > int > int ).

thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
    if_list_int: $o > list_int > list_int > list_int ).

thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_If_001t__Num__Onum,type,
    if_num: $o > num > num > num ).

thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
    if_option_nat: $o > option_nat > option_nat > option_nat ).

thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
    if_option_num: $o > option_num > option_num > option_num ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_If_001t__Rat__Orat,type,
    if_rat: $o > rat > rat > rat ).

thf(sy_c_If_001t__Real__Oreal,type,
    if_real: $o > real > real > real ).

thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
    if_set_int: $o > set_int > set_int > set_int ).

thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
    if_set_nat: $o > set_nat > set_nat > set_nat ).

thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
    if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(sy_c_Int_OAbs__Integ,type,
    abs_Integ: product_prod_nat_nat > int ).

thf(sy_c_Int_ORep__Integ,type,
    rep_Integ: int > product_prod_nat_nat ).

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

thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
    ring_1_Ints_real: set_real ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_List_Odistinct_001t__Int__Oint,type,
    distinct_int: list_int > $o ).

thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
    distinct_nat: list_nat > $o ).

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

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

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

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

thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
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thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).

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

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

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

thf(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
    nth_Code_integer: list_Code_integer > nat > code_integer ).

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

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

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

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o ).

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__Num__Onum_Mt__Num__Onum_J,type,
    nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
    produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o ).

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

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

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

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

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

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

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

thf(sy_c_List_Oremdups_001t__Nat__Onat,type,
    remdups_nat: list_nat > list_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    size_s3445333598471063425nteger: list_Code_integer > nat ).

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
    size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__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__Nat__Onat_J,type,
    size_size_option_nat: option_nat > nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

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

thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
    case_option_int_num: int > ( num > int ) > option_num > int ).

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

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

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

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

thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
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thf(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
    the_num: option_num > num ).

thf(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Code____Numeral__Ointeger_M_062_I_Eo_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Num__Onum_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
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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_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: set_complex > set_complex > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_set_rat: set_rat > set_rat > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Code____Numeral__Ointeger_M_062_I_Eo_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_le211207098394363844omplex: set_complex > set_complex > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
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    produc6406642877701697732et_int: ( num > num > set_int ) > product_prod_num_num > set_int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Nat__Onat_J,type,
    produc1361121860356118632et_nat: ( num > num > set_nat ) > product_prod_num_num > set_nat ).

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

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

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

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

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

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

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

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

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

thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

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

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

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

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

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

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

thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

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

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

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

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

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

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

thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
    real_V1803761363581548252l_real: real > real ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: real > real > real ).

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

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

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

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

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

thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: real > real > $o ).

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

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

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

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

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

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

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

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

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

thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
    suminf_complex: ( nat > complex ) > complex ).

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

thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
    suminf_nat: ( nat > nat ) > nat ).

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

thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
    summable_complex: ( nat > complex ) > $o ).

thf(sy_c_Series_Osummable_001t__Int__Oint,type,
    summable_int: ( nat > int ) > $o ).

thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
    summable_nat: ( nat > nat ) > $o ).

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

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

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

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

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

thf(sy_c_Set_OCollect_001_Eo,type,
    collect_o: ( $o > $o ) > set_o ).

thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
    collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).

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

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

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

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

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

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

thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or890127255671739683et_nat: set_nat > set_set_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__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__Real__Oreal,type,
    topolo4055970368930404560y_real: ( nat > real ) > $o ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
    cot_complex: complex > complex ).

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

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

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

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

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

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

thf(sy_c_Transcendental_Osinh_001t__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_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_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: $o > $o > vEBT_VEBT ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    member1379723562493234055eger_o: produc6271795597528267376eger_o > set_Pr448751882837621926eger_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__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

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

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

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

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

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

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

thf(sy_v_deg____,type,
    deg: nat ).

thf(sy_v_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_sxa____,type,
    sxa: nat ).

thf(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (10111)
thf(fact_0__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
    ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = na ) ).

% \<open>deg div 2 = n\<close>
thf(fact_1__C5_Ohyps_C_I3_J,axiom,
    ! [X: nat,Sx: nat] :
      ( ( ( vEBT_vebt_succ @ summary @ X )
        = ( some_nat @ Sx ) )
      = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X @ Sx ) ) ).

% "5.hyps"(3)
thf(fact_2__C5_Ohyps_C_I9_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% "5.hyps"(9)
thf(fact_3_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_4_option_Oinject,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X2 )
        = ( some_P7363390416028606310at_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_5_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( some_nat @ X2 )
        = ( some_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_6_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( some_num @ X2 )
        = ( some_num @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_7_prod_Oinject,axiom,
    ! [X1: code_integer,X2: $o,Y1: code_integer,Y2: $o] :
      ( ( ( produc6677183202524767010eger_o @ X1 @ X2 )
        = ( produc6677183202524767010eger_o @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_8_prod_Oinject,axiom,
    ! [X1: num,X2: num,Y1: num,Y2: num] :
      ( ( ( product_Pair_num_num @ X1 @ X2 )
        = ( product_Pair_num_num @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_9_prod_Oinject,axiom,
    ! [X1: nat,X2: num,Y1: nat,Y2: num] :
      ( ( ( product_Pair_nat_num @ X1 @ X2 )
        = ( product_Pair_nat_num @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_10_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

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

% prod.inject
thf(fact_12_old_Oprod_Oinject,axiom,
    ! [A: code_integer,B: $o,A2: code_integer,B2: $o] :
      ( ( ( produc6677183202524767010eger_o @ A @ B )
        = ( produc6677183202524767010eger_o @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_13_old_Oprod_Oinject,axiom,
    ! [A: num,B: num,A2: num,B2: num] :
      ( ( ( product_Pair_num_num @ A @ B )
        = ( product_Pair_num_num @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
    ! [A: nat,B: num,A2: nat,B2: num] :
      ( ( ( product_Pair_nat_num @ A @ B )
        = ( product_Pair_nat_num @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

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

% old.prod.inject
thf(fact_17__C5_Ohyps_C_I1_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( ( vEBT_invar_vebt @ X3 @ na )
        & ! [Xa: nat,Xb: nat] :
            ( ( ( vEBT_vebt_succ @ X3 @ Xa )
              = ( some_nat @ Xb ) )
            = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ X3 ) @ Xa @ Xb ) ) ) ) ).

% "5.hyps"(1)
thf(fact_18_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_19__C5_Ohyps_C_I2_J,axiom,
    vEBT_invar_vebt @ summary @ m ).

% "5.hyps"(2)
thf(fact_20_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_21_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K: nat,M: nat] :
          ( X
         != ( product_Pair_nat_nat @ K @ M ) ) ).

% prod_decode_aux.cases
thf(fact_22__C5_Ohyps_C_I6_J,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% "5.hyps"(6)
thf(fact_23__C5_Ohyps_C_I8_J,axiom,
    ( ( mi = ma )
   => ! [X3: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ treeList ) )
       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ).

% "5.hyps"(8)
thf(fact_24_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6271795597528267376eger_o] :
      ~ ! [A3: code_integer,B3: $o] :
          ( Y
         != ( produc6677183202524767010eger_o @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_25_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_num_num] :
      ~ ! [A3: num,B3: num] :
          ( Y
         != ( product_Pair_num_num @ A3 @ B3 ) ) ).

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

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

% old.prod.exhaust
thf(fact_28_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B3: int] :
          ( Y
         != ( product_Pair_int_int @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_29_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( Deg = N ) ) ).

% deg_deg_n
thf(fact_30_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set_nat,X4: nat] :
          ( ( member_nat @ X4 @ Xs )
          & ! [Y3: nat] :
              ( ( member_nat @ Y3 @ Xs )
             => ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ).

% min_in_set_def
thf(fact_31_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set_nat,X4: nat] :
          ( ( member_nat @ X4 @ Xs )
          & ! [Y3: nat] :
              ( ( member_nat @ Y3 @ Xs )
             => ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ) ).

% max_in_set_def
thf(fact_32_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).

% maxbmo
thf(fact_33_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_34__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
    ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).

% \<open>2 \<le> deg\<close>
thf(fact_35_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
             => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
          & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_36_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) ) ) ) ).

% mint_sound
thf(fact_37_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% mint_corr
thf(fact_38_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) ) ) ) ).

% maxt_sound
thf(fact_39_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% maxt_corr
thf(fact_40__C5_Ohyps_C_I5_J,axiom,
    ( m
    = ( suc @ na ) ) ).

% "5.hyps"(5)
thf(fact_41_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y3: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) ) ) ) ).

% lesseq_shift
thf(fact_42_Pair__inject,axiom,
    ! [A: code_integer,B: $o,A2: code_integer,B2: $o] :
      ( ( ( produc6677183202524767010eger_o @ A @ B )
        = ( produc6677183202524767010eger_o @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B = ~ B2 ) ) ) ).

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

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

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

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

% Pair_inject
thf(fact_47_prod__cases,axiom,
    ! [P: produc6271795597528267376eger_o > $o,P2: produc6271795597528267376eger_o] :
      ( ! [A3: code_integer,B3: $o] : ( P @ ( produc6677183202524767010eger_o @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

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

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

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

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

% prod_cases
thf(fact_52_surj__pair,axiom,
    ! [P2: produc6271795597528267376eger_o] :
    ? [X5: code_integer,Y4: $o] :
      ( P2
      = ( produc6677183202524767010eger_o @ X5 @ Y4 ) ) ).

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

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

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

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

% surj_pair
thf(fact_57_add__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% add_shift
thf(fact_58_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: 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 @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_59_add__self__div__2,axiom,
    ! [M2: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = M2 ) ).

% add_self_div_2
thf(fact_60_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_67_Collect__mem__eq,axiom,
    ! [A4: set_int] :
      ( ( collect_int
        @ ^ [X4: int] : ( member_int @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
    ! [A4: set_complex] :
      ( ( collect_complex
        @ ^ [X4: complex] : ( member_complex @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
    ! [A4: set_real] :
      ( ( collect_real
        @ ^ [X4: real] : ( member_real @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
    ! [A4: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
    ! [A4: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_74_Collect__cong,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X5: complex] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_complex @ P )
        = ( collect_complex @ Q ) ) ) ).

% Collect_cong
thf(fact_75_Collect__cong,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X5: real] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_real @ P )
        = ( collect_real @ Q ) ) ) ).

% Collect_cong
thf(fact_76_Collect__cong,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X5: list_nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_list_nat @ P )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_77_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X5: set_nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

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

% Collect_cong
thf(fact_79_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_80_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_81_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).

% mint_corr_help
thf(fact_82_nat__add__left__cancel__le,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_83_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_84_field__sum__of__halves,axiom,
    ! [X: rat] :
      ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = X ) ).

% field_sum_of_halves
thf(fact_85_semiring__norm_I85_J,axiom,
    ! [M2: num] :
      ( ( bit0 @ M2 )
     != one ) ).

% semiring_norm(85)
thf(fact_86_semiring__norm_I83_J,axiom,
    ! [N: num] :
      ( one
     != ( bit0 @ N ) ) ).

% semiring_norm(83)
thf(fact_87_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_88_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_89_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_90_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_91_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_92_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_93_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_94_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_95_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_96_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_97_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N2: nat] :
          ( X
         != ( plus_plus_nat @ N2 @ N2 ) )
     => ~ ! [N2: nat] :
            ( X
           != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).

% even_odd_cases
thf(fact_98_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S ) ) ) ).

% deg_SUcn_Node
thf(fact_99_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
       => ( vEBT_vebt_member @ T @ X ) ) ) ).

% valid_member_both_member_options
thf(fact_100_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
        = ( vEBT_vebt_member @ T @ X ) ) ) ).

% both_member_options_equiv_member
thf(fact_101_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ T @ X )
        = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_102_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numera6690914467698888265omplex @ M2 )
        = ( numera6690914467698888265omplex @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_103_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_real @ M2 )
        = ( numeral_numeral_real @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_104_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_rat @ M2 )
        = ( numeral_numeral_rat @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_105_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_nat @ M2 )
        = ( numeral_numeral_nat @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_106_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_int @ M2 )
        = ( numeral_numeral_int @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_107_semiring__norm_I87_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( bit0 @ M2 )
        = ( bit0 @ N ) )
      = ( M2 = N ) ) ).

% semiring_norm(87)
thf(fact_108_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_109_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_110_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% mint_member
thf(fact_111_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% maxt_member
thf(fact_112_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ X @ Y )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z2 )
              & ( ord_less_nat @ X @ Z2 ) )
           => ( ord_less_eq_nat @ Y @ Z2 ) ) ) ) ).

% succ_member
thf(fact_113_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_114_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_115_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_116_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_117_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_118_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_119_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_120_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_121_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_eq
thf(fact_122_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_123_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_124_Suc__le__mono,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
      = ( ord_less_eq_nat @ N @ M2 ) ) ).

% Suc_le_mono
thf(fact_125_add__Suc__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ M2 @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% add_Suc_right
thf(fact_126_nat__add__left__cancel__less,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_127_semiring__norm_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( bit0 @ ( plus_plus_num @ M2 @ N ) ) ) ).

% semiring_norm(6)
thf(fact_128_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_129_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_130_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y3: nat,X4: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) ) ) ) ).

% greater_shift
thf(fact_131_add__2__eq__Suc_H,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N ) ) ) ).

% add_2_eq_Suc'
thf(fact_132_add__2__eq__Suc,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
      = ( suc @ ( suc @ N ) ) ) ).

% add_2_eq_Suc
thf(fact_133_div2__Suc__Suc,axiom,
    ! [M2: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div2_Suc_Suc
thf(fact_134_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_135__C5_Ohyps_C_I10_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% "5.hyps"(10)
thf(fact_136_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_leI
thf(fact_137_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_eq
thf(fact_138_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ~ ! [Q2: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).

% less_natE
thf(fact_139_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_140_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_141_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_lessD
thf(fact_142_less__add__Suc1,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).

% less_add_Suc1
thf(fact_143_less__add__Suc2,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).

% less_add_Suc2
thf(fact_144_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% le_less_Suc_eq
thf(fact_145_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_Suc_eq_le
thf(fact_146_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_147_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_148_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_149_less__imp__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ? [K: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).

% less_imp_Suc_add
thf(fact_150_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
thf(fact_151_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_152_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_153_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M4: nat] :
              ( ( ord_less_nat @ M4 @ N2 )
              & ~ ( P @ M4 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_154_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N2 )
             => ( P @ M4 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_155_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_156_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K )
               => ( ( P @ I2 @ J2 )
                 => ( ( P @ J2 @ K )
                   => ( P @ I2 @ K ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_157_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K2 )
       => ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).

% less_trans_Suc
thf(fact_158_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( ( ord_less_nat @ S2 @ T )
     => ( S2 != T ) ) ).

% less_not_refl3
thf(fact_159_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( M2 != N ) ) ).

% less_not_refl2
thf(fact_160_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_161_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_SucD
thf(fact_162_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
       => ( M2 = N ) ) ) ).

% less_antisym
thf(fact_163_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M2 )
      = ( ? [M5: nat] :
            ( ( M2
              = ( suc @ M5 ) )
            & ( ord_less_nat @ N @ M5 ) ) ) ) ).

% Suc_less_eq2
thf(fact_164_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ N )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ I3 ) ) ) ) ).

% All_less_Suc
thf(fact_165_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M2 @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).

% not_less_eq
thf(fact_166_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
      = ( ( ord_less_nat @ M2 @ N )
        | ( ord_less_nat @ N @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_167_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_168_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) ) ) ).

% less_Suc_eq
thf(fact_169_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ N )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_170_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_171_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_172_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M2 @ N )
       => ( M2 = N ) ) ) ).

% less_SucE
thf(fact_173_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ( suc @ M2 )
         != N )
       => ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).

% Suc_lessI
thf(fact_174_Suc__lessE,axiom,
    ! [I: nat,K2: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K2 )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K2
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_175_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_lessD
thf(fact_176_Nat_OlessE,axiom,
    ! [I: nat,K2: nat] :
      ( ( ord_less_nat @ I @ K2 )
     => ( ( K2
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K2
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_177_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_real @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_178_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_179_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_num @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_180_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_181_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_182_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_183_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_184_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_185_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_186_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_187_add__One__commute,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ N )
      = ( plus_plus_num @ N @ one ) ) ).

% add_One_commute
thf(fact_188_transitive__stepwise__le,axiom,
    ! [M2: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ! [X5: nat] : ( R @ X5 @ X5 )
       => ( ! [X5: nat,Y4: nat,Z3: nat] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z3 )
               => ( R @ X5 @ Z3 ) ) )
         => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
           => ( R @ M2 @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_189_nat__induct__at__least,axiom,
    ! [M2: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( P @ M2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_190_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
             => ( P @ M4 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_191_not__less__eq__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).

% not_less_eq_eq
thf(fact_192_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_193_le__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M2 @ N )
        | ( M2
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_194_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
     => ? [M: nat] :
          ( M6
          = ( suc @ M ) ) ) ).

% Suc_le_D
thf(fact_195_le__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_196_le__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( M2
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_197_Suc__leD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% Suc_leD
thf(fact_198_add__Suc__shift,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N )
      = ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_199_add__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N )
      = ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% add_Suc
thf(fact_200_nat__arith_Osuc1,axiom,
    ! [A4: nat,K2: nat,A: nat] :
      ( ( A4
        = ( plus_plus_nat @ K2 @ A ) )
     => ( ( suc @ A4 )
        = ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_201_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_202_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( M2 != N )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% le_neq_implies_less
thf(fact_203_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_or_eq_imp_le
thf(fact_204_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_nat @ M3 @ N3 )
          | ( M3 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_205_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_imp_le_nat
thf(fact_206_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
          & ( M3 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_207_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ K2 @ L )
     => ( ( ( plus_plus_nat @ M2 @ L )
          = ( plus_plus_nat @ K2 @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% less_add_eq_less
thf(fact_208_trans__less__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_less_add2
thf(fact_209_trans__less__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_less_add1
thf(fact_210_add__less__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% add_less_mono1
thf(fact_211_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_212_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_213_add__less__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K2 @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_214_add__lessD1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
     => ( ord_less_nat @ I @ K2 ) ) ).

% add_lessD1
thf(fact_215_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).

% Suc_nat_number_of_add
thf(fact_216_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_217_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_218_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_219_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_220_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_221_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_222_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_223_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_224_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_225_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_226_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M2: nat,K2: nat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_nat @ M @ N2 )
         => ( ord_less_nat @ ( F @ M ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_227_Suc__div__le__mono,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_div_le_mono
thf(fact_228_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs: set_nat,X4: nat,Y3: nat] :
          ( ( member_nat @ Y3 @ Xs )
          & ( ord_less_nat @ X4 @ Y3 )
          & ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Xs )
             => ( ( ord_less_nat @ X4 @ Z2 )
               => ( ord_less_eq_nat @ Y3 @ Z2 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_229_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_230_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_231_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_232_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K2: nat,B: nat] :
      ( ( P @ K2 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ? [X5: nat] :
            ( ( P @ X5 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_233_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
      | ( ord_less_eq_nat @ N @ M2 ) ) ).

% nat_le_linear
thf(fact_234_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% le_antisym
thf(fact_235_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% eq_imp_le
thf(fact_236_le__trans,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K2 )
       => ( ord_less_eq_nat @ I @ K2 ) ) ) ).

% le_trans
thf(fact_237_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_238_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_239_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_240_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_241_trans__le__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_le_add2
thf(fact_242_trans__le__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_le_add1
thf(fact_243_add__le__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% add_le_mono1
thf(fact_244_add__le__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K2 @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_245_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ K2 @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K2 @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_246_add__leD2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
     => ( ord_less_eq_nat @ K2 @ N ) ) ).

% add_leD2
thf(fact_247_add__leD1,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% add_leD1
thf(fact_248_le__add2,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).

% le_add2
thf(fact_249_le__add1,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).

% le_add1
thf(fact_250_add__leE,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
     => ~ ( ( ord_less_eq_nat @ M2 @ N )
         => ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).

% add_leE
thf(fact_251_div__le__dividend,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).

% div_le_dividend
thf(fact_252_div__le__mono,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K2 ) @ ( divide_divide_nat @ N @ K2 ) ) ) ).

% div_le_mono
thf(fact_253_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_Bit0
thf(fact_254_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_Bit0
thf(fact_255_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_Bit0
thf(fact_256_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_Bit0
thf(fact_257_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_Bit0
thf(fact_258_divide__numeral__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_259_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_260_divide__numeral__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_261_numeral__Bit0__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N ) ) ).

% numeral_Bit0_div_2
thf(fact_262_numeral__Bit0__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% numeral_Bit0_div_2
thf(fact_263_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y3: nat] : ( vEBT_VEBT_less @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) ) ) ) ).

% less_shift
thf(fact_264_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Mi @ Ma )
        & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).

% mi_ma_2_deg
thf(fact_265_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_266_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( some_nat @ M2 )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% misiz
thf(fact_267_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_268_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_269_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_270_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_271_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_272_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_273_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_274_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_275_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_276_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_277_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_278_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_279_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_280_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_281_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_282_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_283_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_284_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_285_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( ( vEBT_vebt_member @ Tree @ X )
     => ( ( vEBT_invar_vebt @ Tree @ N )
       => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% member_bound
thf(fact_286__C5_Ohyps_C_I7_J,axiom,
    ! [I4: nat] :
      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ X6 ) )
        = ( vEBT_V8194947554948674370ptions @ summary @ I4 ) ) ) ).

% "5.hyps"(7)
thf(fact_287_power__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( power_power_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% power_shift
thf(fact_288_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_289_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_290_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_291_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_292_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_293_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_294_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_295_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_296_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_297_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_298_semiring__norm_I78_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% semiring_norm(78)
thf(fact_299_semiring__norm_I71_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% semiring_norm(71)
thf(fact_300_semiring__norm_I75_J,axiom,
    ! [M2: num] :
      ~ ( ord_less_num @ M2 @ one ) ).

% semiring_norm(75)
thf(fact_301_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_302_semiring__norm_I76_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).

% semiring_norm(76)
thf(fact_303_semiring__norm_I69_J,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M2 ) @ one ) ).

% semiring_norm(69)
thf(fact_304__C5_Ohyps_C_I4_J,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% "5.hyps"(4)
thf(fact_305_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_306_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
      = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_307_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: num > num > num,A: num,B: num] :
      ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
      = ( some_num @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_308_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
      = ( some_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_309_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_310_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_311_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_312_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_313_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( ( plus_plus_real @ I @ K2 )
        = ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_314_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( ( plus_plus_rat @ I @ K2 )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_315_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( ( plus_plus_nat @ I @ K2 )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_316_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( ( plus_plus_int @ I @ K2 )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_317_group__cancel_Oadd1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4
        = ( plus_plus_real @ K2 @ A ) )
     => ( ( plus_plus_real @ A4 @ B )
        = ( plus_plus_real @ K2 @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_318_group__cancel_Oadd1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4
        = ( plus_plus_rat @ K2 @ A ) )
     => ( ( plus_plus_rat @ A4 @ B )
        = ( plus_plus_rat @ K2 @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_319_group__cancel_Oadd1,axiom,
    ! [A4: nat,K2: nat,A: nat,B: nat] :
      ( ( A4
        = ( plus_plus_nat @ K2 @ A ) )
     => ( ( plus_plus_nat @ A4 @ B )
        = ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_320_group__cancel_Oadd1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4
        = ( plus_plus_int @ K2 @ A ) )
     => ( ( plus_plus_int @ A4 @ B )
        = ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_321_group__cancel_Oadd2,axiom,
    ! [B4: real,K2: real,B: real,A: real] :
      ( ( B4
        = ( plus_plus_real @ K2 @ B ) )
     => ( ( plus_plus_real @ A @ B4 )
        = ( plus_plus_real @ K2 @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_322_group__cancel_Oadd2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4
        = ( plus_plus_rat @ K2 @ B ) )
     => ( ( plus_plus_rat @ A @ B4 )
        = ( plus_plus_rat @ K2 @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_323_group__cancel_Oadd2,axiom,
    ! [B4: nat,K2: nat,B: nat,A: nat] :
      ( ( B4
        = ( plus_plus_nat @ K2 @ B ) )
     => ( ( plus_plus_nat @ A @ B4 )
        = ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_324_group__cancel_Oadd2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4
        = ( plus_plus_int @ K2 @ B ) )
     => ( ( plus_plus_int @ A @ B4 )
        = ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_325_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_326_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.assoc
thf(fact_327_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_328_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_329_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_330_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_331_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_332_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_333_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_334_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_335_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A5: real,B5: real] : ( plus_plus_real @ B5 @ A5 ) ) ) ).

% add.commute
thf(fact_336_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A5: rat,B5: rat] : ( plus_plus_rat @ B5 @ A5 ) ) ) ).

% add.commute
thf(fact_337_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B5: nat] : ( plus_plus_nat @ B5 @ A5 ) ) ) ).

% add.commute
thf(fact_338_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A5: int,B5: int] : ( plus_plus_int @ B5 @ A5 ) ) ) ).

% add.commute
thf(fact_339_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_340_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_341_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_342_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_343_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_344_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_345_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_346_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_347_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_348_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_349_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_350_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_351_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_352_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_353_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_354_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_355_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_eq_real @ K2 @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_356_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K2 @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_357_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_358_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K2 @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_359_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_eq_real @ K2 @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_360_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K2 @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_361_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_362_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K2 @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_363_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_364_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_365_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_366_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_367_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_368_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_369_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_370_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_371_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( B
           != ( plus_plus_nat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_372_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_373_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_374_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_375_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_376_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] :
        ? [C3: nat] :
          ( B5
          = ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_377_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_378_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_379_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_380_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_381_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_382_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_383_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_384_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_385_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_real @ K2 @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_386_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_387_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_388_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_389_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_real @ K2 @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_390_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_391_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_392_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_393_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_394_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_395_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_396_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_397_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_398_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_399_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_400_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_401_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_402_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_403_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_404_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_405_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_406_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_407_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_408_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_409_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_410_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_411_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_412_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_413_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_414_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_415_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_416_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_417_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_real @ K2 @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_418_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_419_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_420_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_421_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_eq_real @ K2 @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_422_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_423_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_424_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_425_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_426_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_427_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_428_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_429_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_430_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_431_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_432_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_433_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M: nat] :
              ( ( ( some_nat @ M )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_434__092_060open_062length_AtreeList_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_Am_092_060close_062,axiom,
    ( ( ( size_s6755466524823107622T_VEBT @ treeList )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
    & ( vEBT_invar_vebt @ summary @ m ) ) ).

% \<open>length treeList = 2 ^ m \<and> invar_vebt summary m\<close>
thf(fact_435_div__exp__eq,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).

% div_exp_eq
thf(fact_436_div__exp__eq,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).

% div_exp_eq
thf(fact_437__C5_Ohyps_C_I11_J,axiom,
    ( ( mi != ma )
   => ! [I4: nat] :
        ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ( ( vEBT_VEBT_high @ ma @ na )
              = I4 )
           => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
          & ! [X3: nat] :
              ( ( ( ( vEBT_VEBT_high @ X3 @ na )
                  = I4 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ X3 @ na ) ) )
             => ( ( ord_less_nat @ mi @ X3 )
                & ( ord_less_eq_nat @ X3 @ ma ) ) ) ) ) ) ).

% "5.hyps"(11)
thf(fact_438_self__le__ge2__pow,axiom,
    ! [K2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
     => ( ord_less_eq_nat @ M2 @ ( power_power_nat @ K2 @ M2 ) ) ) ).

% self_le_ge2_pow
thf(fact_439_power2__nat__le__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% power2_nat_le_eq_le
thf(fact_440_power2__nat__le__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% power2_nat_le_imp_le
thf(fact_441_enat__ord__number_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(1)
thf(fact_442_less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% less_exp
thf(fact_443_enat__ord__number_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(2)
thf(fact_444_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
        = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% power_minus_is_div
thf(fact_445_inthall,axiom,
    ! [Xs2: list_real,P: real > $o,N: nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_real2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
       => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_446_inthall,axiom,
    ! [Xs2: list_complex,P: complex > $o,N: nat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ ( set_complex2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
       => ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_447_inthall,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,N: nat] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
       => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_448_inthall,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_449_inthall,axiom,
    ! [Xs2: list_o,P: $o > $o,N: nat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
       => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_450_inthall,axiom,
    ! [Xs2: list_nat,P: nat > $o,N: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_451_inthall,axiom,
    ! [Xs2: list_int,P: int > $o,N: nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
       => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_452_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_453_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X4: nat,N3: nat] : ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% high_def
thf(fact_454_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% high_bound_aux
thf(fact_455_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_456_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_457_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_458_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_459_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_460_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_461_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_462_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_463_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_464_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_465_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_466_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_467_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_468_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_469_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_470_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_471_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_472_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_473_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_474_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_475_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_476_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_477_Suc__diff__diff,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K2 ) ) ).

% Suc_diff_diff
thf(fact_478_diff__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% diff_Suc_Suc
thf(fact_479_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_480_diff__diff__left,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% diff_diff_left
thf(fact_481_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_482_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_483_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).

% Nat.add_diff_assoc
thf(fact_484_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ Summary ) @ X ) ) ) ) ).

% both_member_options_ding
thf(fact_485_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ 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 @ TreeList ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( 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_486_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_487_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_488_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M4: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M4 @ N2 )
             => ( P @ M4 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% enat_less_induct
thf(fact_489_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_490_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_491_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_492_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_493_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_494_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_495_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_496_diff__commute,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).

% diff_commute
thf(fact_497_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_498_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y: list_o] :
      ( ( ( size_size_list_o @ X )
       != ( size_size_list_o @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_499_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_500_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_501_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_502_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_503_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_504_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_505_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_506_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_507_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_508_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_509_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_510_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_511_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_512_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_513_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_514_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_515_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_516_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_517_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_518_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_519_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_520_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_521_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_522_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_523_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_524_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_525_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_526_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_527_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
      = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).

% add_diff_add
thf(fact_528_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_529_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_530_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_531_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_532_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_533_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_534_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ( plus_plus_rat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_rat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_535_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_536_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_537_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_538_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_539_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_540_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_541_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_542_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_543_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_544_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_545_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_546_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_547_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_548_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_549_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_550_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A
        = ( minus_minus_rat @ C @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_551_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_552_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_553_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = C )
      = ( A
        = ( plus_plus_rat @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_554_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_555_group__cancel_Osub1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4
        = ( plus_plus_real @ K2 @ A ) )
     => ( ( minus_minus_real @ A4 @ B )
        = ( plus_plus_real @ K2 @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_556_group__cancel_Osub1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4
        = ( plus_plus_rat @ K2 @ A ) )
     => ( ( minus_minus_rat @ A4 @ B )
        = ( plus_plus_rat @ K2 @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_557_group__cancel_Osub1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4
        = ( plus_plus_int @ K2 @ A ) )
     => ( ( minus_minus_int @ A4 @ B )
        = ( plus_plus_int @ K2 @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_558_zero__induct__lemma,axiom,
    ! [P: nat > $o,K2: nat,I: nat] :
      ( ( P @ K2 )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_559_less__imp__diff__less,axiom,
    ! [J: nat,K2: nat,N: nat] :
      ( ( ord_less_nat @ J @ K2 )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).

% less_imp_diff_less
thf(fact_560_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ord_less_nat @ M2 @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_561_diff__le__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_562_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_563_diff__le__self,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).

% diff_le_self
thf(fact_564_diff__le__mono,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_565_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_566_le__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% le_diff_iff
thf(fact_567_eq__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ( minus_minus_nat @ M2 @ K2 )
            = ( minus_minus_nat @ N @ K2 ) )
          = ( M2 = N ) ) ) ) ).

% eq_diff_iff
thf(fact_568_diff__add__inverse2,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
      = M2 ) ).

% diff_add_inverse2
thf(fact_569_diff__add__inverse,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
      = M2 ) ).

% diff_add_inverse
thf(fact_570_diff__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% diff_cancel2
thf(fact_571_Nat_Odiff__cancel,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% Nat.diff_cancel
thf(fact_572_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_573_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_574_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_575_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_576_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_577_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_578_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_579_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_580_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_581_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_582_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_583_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_584_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_585_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_586_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_587_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_588_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_589_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_590_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_591_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_592_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_593_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_594_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).

% diff_less_Suc
thf(fact_595_Suc__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M2 @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_596_Suc__diff__le,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_597_less__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( ord_less_nat @ M2 @ N ) ) ) ) ).

% less_diff_iff
thf(fact_598_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_599_less__diff__conv,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).

% less_diff_conv
thf(fact_600_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ ( ord_less_nat @ M2 @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_601_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K2 )
        = ( J
          = ( plus_plus_nat @ K2 @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_602_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_603_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_604_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_605_le__diff__conv,axiom,
    ! [J: nat,K2: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).

% le_diff_conv
thf(fact_606_power2__commute,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ ( minus_minus_complex @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_607_power2__commute,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_608_power2__commute,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ ( minus_minus_rat @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_609_power2__commute,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_610_less__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).

% less_diff_conv2
thf(fact_611_diff__le__diff__pow,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K2 @ M2 ) @ ( power_power_nat @ K2 @ N ) ) ) ) ).

% diff_le_diff_pow
thf(fact_612_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2 = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M2 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X5 )
                                      & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_613_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M2 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X5 )
                                      & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_614_power__divide,axiom,
    ! [A: complex,B: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
      = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).

% power_divide
thf(fact_615_power__divide,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
      = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).

% power_divide
thf(fact_616_power__divide,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
      = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).

% power_divide
thf(fact_617_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N3: nat,TreeList3: list_VEBT_VEBT,X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X4 @ N3 ) ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) ) ) ).

% in_children_def
thf(fact_618_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: 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 @ TreeList ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ Summary ) @ X ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_619_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: 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 @ TreeList @ Summary ) @ X )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( 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_620_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi != Ma )
       => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
          = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% summaxma
thf(fact_621_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_622_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M2: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
       => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).

% set_n_deg_not_0
thf(fact_623_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_624_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_625_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_626_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_627_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_628_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_629_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_630_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_631_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
        = X ) ) ).

% low_inv
thf(fact_632_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
        = Y ) ) ).

% high_inv
thf(fact_633_nth__mem,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_634_nth__mem,axiom,
    ! [N: nat,Xs2: list_complex] :
      ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_635_nth__mem,axiom,
    ! [N: nat,Xs2: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
     => ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).

% nth_mem
thf(fact_636_nth__mem,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_637_nth__mem,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_638_nth__mem,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_639_nth__mem,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_640__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
    ord_less_eq_nat @ one_one_nat @ na ).

% \<open>1 \<le> n\<close>
thf(fact_641_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_642_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_643_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_644_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_645_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_646_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_647_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_648_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_649_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_650_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_651_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_652_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_653_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_654_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_655_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_656_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_657_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_658_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_659_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_660_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_661_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_662_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_663_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_664_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_665_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_666_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_667_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_668_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ one_one_rat @ N )
      = one_one_rat ) ).

% power_one
thf(fact_669_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_670_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_real @ one_one_real @ N )
      = one_one_real ) ).

% power_one
thf(fact_671_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_672_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ one_one_complex @ N )
      = one_one_complex ) ).

% power_one
thf(fact_673_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L2: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L2 ) ) ) ).

% bit_concat_def
thf(fact_674_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_675_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_676_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_677_power__one__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_678_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( ! [Y3: product_prod_nat_nat] :
            ( X
           != ( some_P7363390416028606310at_nat @ Y3 ) ) )
      = ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_679_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ( ! [Y3: nat] :
            ( X
           != ( some_nat @ Y3 ) ) )
      = ( X = none_nat ) ) ).

% not_Some_eq
thf(fact_680_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ( ! [Y3: num] :
            ( X
           != ( some_num @ Y3 ) ) )
      = ( X = none_num ) ) ).

% not_Some_eq
thf(fact_681_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
      = ( ? [Y3: product_prod_nat_nat] :
            ( X
            = ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_682_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
      = ( ? [Y3: nat] :
            ( X
            = ( some_nat @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_683_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
      = ( ? [Y3: num] :
            ( X
            = ( some_num @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_684_nat__mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ N )
        = one_one_nat )
      = ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_685_nat__1__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M2 @ N ) )
      = ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_686_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_687_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_688_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_689_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_690_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_691_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_692_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_693_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_694_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_695_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_696_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_697_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_698_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_699_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_700_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V: num] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_701_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_702_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_703_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_704_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6690914467698888265omplex @ N )
        = one_one_complex )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_705_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_real @ N )
        = one_one_real )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_706_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_rat @ N )
        = one_one_rat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_707_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_708_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_709_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_710_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_711_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_712_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_713_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_714_power__inject__exp,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M2 )
          = ( power_power_real @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_715_power__inject__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M2 )
          = ( power_power_rat @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_716_power__inject__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M2 )
          = ( power_power_nat @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_717_power__inject__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M2 )
          = ( power_power_int @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_718_mult__Suc__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( times_times_nat @ M2 @ ( suc @ N ) )
      = ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N ) ) ) ).

% mult_Suc_right
thf(fact_719_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_720_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_721_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_722_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_723_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_724_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_725_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_726_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_727_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_728_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_729_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_730_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_731_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_732_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_733_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_734_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_735_power__add__numeral2,axiom,
    ! [A: complex,M2: num,N: num,B: complex] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_736_power__add__numeral2,axiom,
    ! [A: real,M2: num,N: num,B: real] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_737_power__add__numeral2,axiom,
    ! [A: rat,M2: num,N: num,B: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_738_power__add__numeral2,axiom,
    ! [A: nat,M2: num,N: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_739_power__add__numeral2,axiom,
    ! [A: int,M2: num,N: num,B: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_740_power__add__numeral,axiom,
    ! [A: complex,M2: num,N: num] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_741_power__add__numeral,axiom,
    ! [A: real,M2: num,N: num] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_742_power__add__numeral,axiom,
    ! [A: rat,M2: num,N: num] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_743_power__add__numeral,axiom,
    ! [A: nat,M2: num,N: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_744_power__add__numeral,axiom,
    ! [A: int,M2: num,N: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_745_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_746_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_747_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_748_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_749_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_750_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_751_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_752_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_753_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_754_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_755_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( N
          = ( suc @ ( suc @ Va ) ) )
       => ( ~ ( ord_less_nat @ Ma @ Mi )
         => ( ( Ma != Mi )
           => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).

% nested_mint
thf(fact_756_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_757_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_758_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_759_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_760_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_761_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_762_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_763_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_764_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_765_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_766_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_767_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_768_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_769_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_770_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_771_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_772_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_773_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_774_less__1__mult,axiom,
    ! [M2: real,N: real] :
      ( ( ord_less_real @ one_one_real @ M2 )
     => ( ( ord_less_real @ one_one_real @ N )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_775_less__1__mult,axiom,
    ! [M2: rat,N: rat] :
      ( ( ord_less_rat @ one_one_rat @ M2 )
     => ( ( ord_less_rat @ one_one_rat @ N )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_776_less__1__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M2 )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_777_less__1__mult,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M2 )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_778_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_779_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_780_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
        = ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( ( the_nat @ Option )
              = ( the_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_781_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
        = ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( ( the_Pr8591224930841456533at_nat @ Option )
              = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_782_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
        = ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( ( the_num @ Option )
              = ( the_num @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_783_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
      = ( X = one_one_complex ) ) ).

% one_reorient
thf(fact_784_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_785_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_786_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_787_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_788_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_789_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_790_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_791_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_792_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_793_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_794_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_795_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_796_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_797_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A5: real,B5: real] : ( times_times_real @ B5 @ A5 ) ) ) ).

% mult.commute
thf(fact_798_mult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A5: rat,B5: rat] : ( times_times_rat @ B5 @ A5 ) ) ) ).

% mult.commute
thf(fact_799_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A5: nat,B5: nat] : ( times_times_nat @ B5 @ A5 ) ) ) ).

% mult.commute
thf(fact_800_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A5: int,B5: int] : ( times_times_int @ B5 @ A5 ) ) ) ).

% mult.commute
thf(fact_801_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_802_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_803_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_804_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_805_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_806_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_807_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_808_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_809_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_810_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_811_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_812_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_813_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_814_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_815_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X @ Y )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_816_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_817_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_818_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_819_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_820_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_821_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_822_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_823_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_824_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_825_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_826_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_827_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_828_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_829_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_830_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_831_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_832_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_833_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% distrib_left
thf(fact_834_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_835_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_836_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_837_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% distrib_right
thf(fact_838_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_839_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_840_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_841_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_842_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_843_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_844_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_845_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_846_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_847_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_848_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_849_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_850_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_851_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
      = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_852_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_853_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_854_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_855_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_856_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_857_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_858_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_859_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_860_square__diff__one__factored,axiom,
    ! [X: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_861_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_862_power__less__power__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_863_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_864_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_865_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_866_power__gt1__lemma,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_867_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_868_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_869_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_870_power__commutes,axiom,
    ! [A: complex,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).

% power_commutes
thf(fact_871_power__commutes,axiom,
    ! [A: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
      = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).

% power_commutes
thf(fact_872_power__commutes,axiom,
    ! [A: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).

% power_commutes
thf(fact_873_power__commutes,axiom,
    ! [A: nat,N: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_commutes
thf(fact_874_power__commutes,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_commutes
thf(fact_875_power__mult__distrib,axiom,
    ! [A: complex,B: complex,N: nat] :
      ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
      = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_876_power__mult__distrib,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_877_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_878_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_879_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_880_power__commuting__commutes,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y )
        = ( times_times_complex @ Y @ ( power_power_complex @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_881_power__commuting__commutes,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
        = ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_882_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = ( times_times_rat @ Y @ X ) )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y )
        = ( times_times_rat @ Y @ ( power_power_rat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_883_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = ( times_times_nat @ Y @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_884_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = ( times_times_int @ Y @ X ) )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_885_Suc__mult__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ( times_times_nat @ ( suc @ K2 ) @ M2 )
        = ( times_times_nat @ ( suc @ K2 ) @ N ) )
      = ( M2 = N ) ) ).

% Suc_mult_cancel1
thf(fact_886_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc5542196010084753463at_nat] :
      ( ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
          ( X
         != ( produc2899441246263362727at_nat @ Uu @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv ) ) )
     => ( ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
            ( X
           != ( produc2899441246263362727at_nat @ Uw @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( X
             != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_887_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc8306885398267862888on_nat] :
      ( ! [Uu: nat > nat > nat,Uv: option_nat] :
          ( X
         != ( produc8929957630744042906on_nat @ Uu @ ( produc5098337634421038937on_nat @ none_nat @ Uv ) ) )
     => ( ! [Uw: nat > nat > nat,V2: nat] :
            ( X
           != ( produc8929957630744042906on_nat @ Uw @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > nat,A3: nat,B3: nat] :
              ( X
             != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_888_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc1193250871479095198on_num] :
      ( ! [Uu: num > num > num,Uv: option_num] :
          ( X
         != ( produc5778274026573060048on_num @ Uu @ ( produc8585076106096196333on_num @ none_num @ Uv ) ) )
     => ( ! [Uw: num > num > num,V2: num] :
            ( X
           != ( produc5778274026573060048on_num @ Uw @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > num,A3: num,B3: num] :
              ( X
             != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_889_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc5491161045314408544at_nat] :
      ( ! [Uu: product_prod_nat_nat > product_prod_nat_nat > $o,Uv: option4927543243414619207at_nat] :
          ( X
         != ( produc3994169339658061776at_nat @ Uu @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv ) ) )
     => ( ! [Uw: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
            ( X
           != ( produc3994169339658061776at_nat @ Uw @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y4: product_prod_nat_nat] :
              ( X
             != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X5 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_890_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc2233624965454879586on_nat] :
      ( ! [Uu: nat > nat > $o,Uv: option_nat] :
          ( X
         != ( produc4035269172776083154on_nat @ Uu @ ( produc5098337634421038937on_nat @ none_nat @ Uv ) ) )
     => ( ! [Uw: nat > nat > $o,V2: nat] :
            ( X
           != ( produc4035269172776083154on_nat @ Uw @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > $o,X5: nat,Y4: nat] :
              ( X
             != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X5 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_891_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc7036089656553540234on_num] :
      ( ! [Uu: num > num > $o,Uv: option_num] :
          ( X
         != ( produc3576312749637752826on_num @ Uu @ ( produc8585076106096196333on_num @ none_num @ Uv ) ) )
     => ( ! [Uw: num > num > $o,V2: num] :
            ( X
           != ( produc3576312749637752826on_num @ Uw @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > $o,X5: num,Y4: num] :
              ( X
             != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X5 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_892_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_893_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: product_prod_nat_nat,B3: nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_894_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A3: product_prod_nat_nat,B3: num] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_895_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: nat,B3: product_prod_nat_nat] :
              ( ( X
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_896_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: nat,B3: nat] :
              ( ( X
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_897_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A3: nat,B3: num] :
              ( ( X
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_898_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: num,B3: product_prod_nat_nat] :
              ( ( X
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_899_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: num,B3: nat] :
              ( ( X
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_900_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A3: num,B3: num] :
              ( ( X
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_901_split__option__all,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ! [X7: option4927543243414619207at_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          & ! [X4: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_902_split__option__all,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ! [X7: option_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          & ! [X4: nat] : ( P4 @ ( some_nat @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_903_split__option__all,axiom,
    ( ( ^ [P3: option_num > $o] :
        ! [X7: option_num] : ( P3 @ X7 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          & ! [X4: num] : ( P4 @ ( some_num @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_904_split__option__ex,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ? [X7: option4927543243414619207at_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          | ? [X4: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_905_split__option__ex,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ? [X7: option_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          | ? [X4: nat] : ( P4 @ ( some_nat @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_906_split__option__ex,axiom,
    ( ( ^ [P3: option_num > $o] :
        ? [X7: option_num] : ( P3 @ X7 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          | ? [X4: num] : ( P4 @ ( some_num @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_907_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X22: product_prod_nat_nat] :
            ( Y
           != ( some_P7363390416028606310at_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_908_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X22: nat] :
            ( Y
           != ( some_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_909_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X22: num] :
            ( Y
           != ( some_num @ X22 ) ) ) ).

% option.exhaust
thf(fact_910_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X2 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_911_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option
        = ( some_nat @ X2 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_912_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option
        = ( some_num @ X2 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_913_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_914_option_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( none_nat
     != ( some_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_915_option_Odistinct_I1_J,axiom,
    ! [X2: num] :
      ( none_num
     != ( some_num @ X2 ) ) ).

% option.distinct(1)
thf(fact_916_power__mult,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ M2 @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A @ M2 ) @ N ) ) ).

% power_mult
thf(fact_917_power__mult,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ M2 @ N ) )
      = ( power_power_real @ ( power_power_real @ A @ M2 ) @ N ) ) ).

% power_mult
thf(fact_918_power__mult,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ M2 @ N ) )
      = ( power_power_int @ ( power_power_int @ A @ M2 ) @ N ) ) ).

% power_mult
thf(fact_919_power__mult,axiom,
    ! [A: complex,M2: nat,N: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ M2 @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A @ M2 ) @ N ) ) ).

% power_mult
thf(fact_920_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_921_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_922_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_923_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_924_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_925_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_926_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_927_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_928_le__cube,axiom,
    ! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).

% le_cube
thf(fact_929_le__square,axiom,
    ! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).

% le_square
thf(fact_930_mult__le__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K2 @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_931_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).

% mult_le_mono1
thf(fact_932_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ).

% mult_le_mono2
thf(fact_933_add__mult__distrib,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K2 )
      = ( plus_plus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).

% add_mult_distrib
thf(fact_934_add__mult__distrib2,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( times_times_nat @ K2 @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) ) ) ).

% add_mult_distrib2
thf(fact_935_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K2 ) ) ).

% left_add_mult_distrib
thf(fact_936_diff__mult__distrib2,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( times_times_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_937_diff__mult__distrib,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K2 )
      = ( minus_minus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).

% diff_mult_distrib
thf(fact_938_div__mult2__eq,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( divide_divide_nat @ M2 @ ( times_times_nat @ N @ Q3 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M2 @ N ) @ Q3 ) ) ).

% div_mult2_eq
thf(fact_939_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_940_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_941_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_942_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_943_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_944_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_945_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_946_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
      = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_947_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_948_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_949_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_950_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_951_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_952_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_953_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_954_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_955_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_956_option_Osel,axiom,
    ! [X2: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_957_option_Osel,axiom,
    ! [X2: nat] :
      ( ( the_nat @ ( some_nat @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_958_option_Osel,axiom,
    ! [X2: num] :
      ( ( the_num @ ( some_num @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_959_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y )
     => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_960_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uu2 @ none_P5556105721700978146at_nat @ Uv2 )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_961_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: num > num > num,Uv2: option_num] :
      ( ( vEBT_V819420779217536731ft_num @ Uu2 @ none_num @ Uv2 )
      = none_num ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_962_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: nat > nat > nat,Uv2: option_nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uu2 @ none_nat @ Uv2 )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_963_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_964_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_965_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_966_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_967_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_968_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_969_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_970_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_971_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_972_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_973_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_974_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_975_mult__numeral__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_976_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_977_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_978_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_979_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_980_mult__numeral__1__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_981_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_982_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_983_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_984_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_985_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_986_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_987_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_988_power__Suc2,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_989_power__Suc2,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_990_power__Suc2,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_991_power__Suc2,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_992_power__Suc2,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_993_power__Suc,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ N ) )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).

% power_Suc
thf(fact_994_power__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ N ) )
      = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).

% power_Suc
thf(fact_995_power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).

% power_Suc
thf(fact_996_power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_Suc
thf(fact_997_power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_Suc
thf(fact_998_Suc__mult__less__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_mult_less_cancel1
thf(fact_999_power__add,axiom,
    ! [A: complex,M2: nat,N: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_add
thf(fact_1000_power__add,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) ) ) ).

% power_add
thf(fact_1001_power__add,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_add
thf(fact_1002_power__add,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ).

% power_add
thf(fact_1003_power__add,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ).

% power_add
thf(fact_1004_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).

% one_le_numeral
thf(fact_1005_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_1006_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_1007_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_1008_Suc__mult__le__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% Suc_mult_le_cancel1
thf(fact_1009_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_1010_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_1011_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_1012_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_1013_mult__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ M2 ) @ N )
      = ( plus_plus_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ).

% mult_Suc
thf(fact_1014_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_1015_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_1016_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_1017_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_1018_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_1019_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_1020_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_1021_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_1022_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_1023_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_1024_one__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).

% one_le_power
thf(fact_1025_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).

% one_le_power
thf(fact_1026_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).

% one_le_power
thf(fact_1027_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).

% one_le_power
thf(fact_1028_less__mult__imp__div__less,axiom,
    ! [M2: nat,I: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_1029_power__one__over,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).

% power_one_over
thf(fact_1030_power__one__over,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).

% power_one_over
thf(fact_1031_power__one__over,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).

% power_one_over
thf(fact_1032_div__times__less__eq__dividend,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) @ M2 ) ).

% div_times_less_eq_dividend
thf(fact_1033_times__div__less__eq__dividend,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) @ M2 ) ).

% times_div_less_eq_dividend
thf(fact_1034_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_1035_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_1036_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_1037_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_1038_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_1039_power__odd__eq,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1040_power__odd__eq,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1041_power__odd__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1042_power__odd__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1043_power__odd__eq,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1044_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb2 )
        = Y )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] :
                ( Xa2
                = ( some_P7363390416028606310at_nat @ V2 ) )
           => ( ( Xb2 = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xa2
                  = ( some_P7363390416028606310at_nat @ A3 ) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xb2
                      = ( some_P7363390416028606310at_nat @ B3 ) )
                   => ( Y
                     != ( some_P7363390416028606310at_nat @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1045_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: num > num > num,Xa2: option_num,Xb2: option_num,Y: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb2 )
        = Y )
     => ( ( ( Xa2 = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] :
                ( Xa2
                = ( some_num @ V2 ) )
           => ( ( Xb2 = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A3: num] :
                ( ( Xa2
                  = ( some_num @ A3 ) )
               => ! [B3: num] :
                    ( ( Xb2
                      = ( some_num @ B3 ) )
                   => ( Y
                     != ( some_num @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1046_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb2 )
        = Y )
     => ( ( ( Xa2 = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] :
                ( Xa2
                = ( some_nat @ V2 ) )
           => ( ( Xb2 = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xa2
                  = ( some_nat @ A3 ) )
               => ! [B3: nat] :
                    ( ( Xb2
                      = ( some_nat @ B3 ) )
                   => ( Y
                     != ( some_nat @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1047_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uw2 @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1048_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: num > num > num,V: num] :
      ( ( vEBT_V819420779217536731ft_num @ Uw2 @ ( some_num @ V ) @ none_num )
      = none_num ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1049_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: nat > nat > nat,V: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uw2 @ ( some_nat @ V ) @ none_nat )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1050_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_1051_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1052_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_1053_power__gt1,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_1054_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_1055_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_1056_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_1057_power__less__imp__less__exp,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1058_power__less__imp__less__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1059_power__less__imp__less__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1060_power__less__imp__less__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1061_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_1062_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_1063_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_1064_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_1065_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_1066_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M2
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_1067_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_1068_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_1069_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_1070_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_1071_power__increasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_1072_power__increasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_1073_power__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_1074_power__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_1075_subset__code_I1_J,axiom,
    ! [Xs2: list_real,B4: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B4 )
      = ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
           => ( member_real @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1076_subset__code_I1_J,axiom,
    ! [Xs2: list_complex,B4: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B4 )
      = ( ! [X4: complex] :
            ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
           => ( member_complex @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1077_subset__code_I1_J,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ B4 )
      = ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
           => ( member8440522571783428010at_nat @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1078_subset__code_I1_J,axiom,
    ! [Xs2: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B4 )
      = ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( member_VEBT_VEBT @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1079_subset__code_I1_J,axiom,
    ! [Xs2: list_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B4 )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
           => ( member_int @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1080_subset__code_I1_J,axiom,
    ! [Xs2: list_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B4 )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
           => ( member_nat @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_1081_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1082_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1083_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1084_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_int] :
      ( ( size_size_list_int @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1085_neq__if__length__neq,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
       != ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1086_neq__if__length__neq,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
       != ( size_size_list_o @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1087_neq__if__length__neq,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1088_neq__if__length__neq,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
       != ( size_size_list_int @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1089_mult__2,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2
thf(fact_1090_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_1091_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_1092_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_1093_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_1094_mult__2__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2_right
thf(fact_1095_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_1096_mult__2__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_1097_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_1098_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_1099_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_1100_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_1101_left__add__twice,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_1102_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_1103_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_1104_power4__eq__xxxx,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_1105_power4__eq__xxxx,axiom,
    ! [X: real] :
      ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_1106_power4__eq__xxxx,axiom,
    ! [X: rat] :
      ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_1107_power4__eq__xxxx,axiom,
    ! [X: nat] :
      ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_1108_power4__eq__xxxx,axiom,
    ! [X: int] :
      ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_1109_power2__eq__square,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_complex @ A @ A ) ) ).

% power2_eq_square
thf(fact_1110_power2__eq__square,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ A @ A ) ) ).

% power2_eq_square
thf(fact_1111_power2__eq__square,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_rat @ A @ A ) ) ).

% power2_eq_square
thf(fact_1112_power2__eq__square,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ A @ A ) ) ).

% power2_eq_square
thf(fact_1113_power2__eq__square,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_int @ A @ A ) ) ).

% power2_eq_square
thf(fact_1114_power__even__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_1115_power__even__eq,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_1116_power__even__eq,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_1117_power__even__eq,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_1118_div__nat__eqI,axiom,
    ! [N: nat,Q3: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M2 )
     => ( ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
       => ( ( divide_divide_nat @ M2 @ N )
          = Q3 ) ) ) ).

% div_nat_eqI
thf(fact_1119_power__le__imp__le__exp,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_1120_power__le__imp__le__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_1121_power__le__imp__le__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_1122_power__le__imp__le__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_1123_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_1124_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_1125_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_1126_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_1127_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_1128_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_1129_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_1130_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_1131_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_1132_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_1133_power2__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_1134_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_1135_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_1136_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1137_length__induct,axiom,
    ! [P: list_o > $o,Xs2: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys2: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1138_length__induct,axiom,
    ! [P: list_nat > $o,Xs2: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1139_length__induct,axiom,
    ! [P: list_int > $o,Xs2: list_int] :
      ( ! [Xs3: list_int] :
          ( ! [Ys2: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1140_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_1141_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_1142_power2__diff,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_1143_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_1144_ex__power__ivl1,axiom,
    ! [B: nat,K2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K2 )
       => ? [N2: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K2 )
            & ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_1145_ex__power__ivl2,axiom,
    ! [B: nat,K2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
       => ? [N2: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K2 )
            & ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_1146_add__le__imp__le__diff,axiom,
    ! [I: real,K2: real,N: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ N )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_1147_add__le__imp__le__diff,axiom,
    ! [I: rat,K2: rat,N: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ N )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_1148_add__le__imp__le__diff,axiom,
    ! [I: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_1149_add__le__imp__le__diff,axiom,
    ! [I: int,K2: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_1150_add__le__add__imp__diff__le,axiom,
    ! [I: real,K2: real,N: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ N )
     => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K2 ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ N )
         => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K2 ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N @ K2 ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1151_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K2: rat,N: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ N )
     => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K2 ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ N )
         => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K2 ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K2 ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1152_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K2: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1153_add__le__add__imp__diff__le,axiom,
    ! [I: int,K2: int,N: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K2 ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1154_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_1155_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ ( ord_less_rat @ A @ B )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1156_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_1157_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_1158_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_VEBT_VEBT,Z4: list_VEBT_VEBT] : Y6 = Z4 )
    = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( nth_VEBT_VEBT @ Xs @ I3 )
                = ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1159_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_o,Z4: list_o] : Y6 = Z4 )
    = ( ^ [Xs: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
             => ( ( nth_o @ Xs @ I3 )
                = ( nth_o @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1160_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_nat,Z4: list_nat] : Y6 = Z4 )
    = ( ^ [Xs: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
             => ( ( nth_nat @ Xs @ I3 )
                = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1161_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_int,Z4: list_int] : Y6 = Z4 )
    = ( ^ [Xs: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
             => ( ( nth_int @ Xs @ I3 )
                = ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1162_Skolem__list__nth,axiom,
    ! [K2: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K2 )
           => ? [X6: vEBT_VEBT] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
              = K2 )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K2 )
               => ( P @ I3 @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1163_Skolem__list__nth,axiom,
    ! [K2: nat,P: nat > $o > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K2 )
           => ? [X6: $o] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs: list_o] :
            ( ( ( size_size_list_o @ Xs )
              = K2 )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K2 )
               => ( P @ I3 @ ( nth_o @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1164_Skolem__list__nth,axiom,
    ! [K2: nat,P: nat > nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K2 )
           => ? [X6: nat] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K2 )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K2 )
               => ( P @ I3 @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1165_Skolem__list__nth,axiom,
    ! [K2: nat,P: nat > int > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K2 )
           => ? [X6: int] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs: list_int] :
            ( ( ( size_size_list_int @ Xs )
              = K2 )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K2 )
               => ( P @ I3 @ ( nth_int @ Xs @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1166_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
              = ( nth_VEBT_VEBT @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1167_nth__equalityI,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
           => ( ( nth_o @ Xs2 @ I2 )
              = ( nth_o @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1168_nth__equalityI,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1169_nth__equalityI,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
           => ( ( nth_int @ Xs2 @ I2 )
              = ( nth_int @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1170_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1171_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P: $o > $o] :
      ( ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
           => ( P @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
           => ( P @ ( nth_o @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1172_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1173_all__set__conv__all__nth,axiom,
    ! [Xs2: list_int,P: int > $o] :
      ( ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
           => ( P @ X4 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
           => ( P @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1174_all__nth__imp__all__set,axiom,
    ! [Xs2: list_real,P: real > $o,X: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
         => ( P @ ( nth_real @ Xs2 @ I2 ) ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1175_all__nth__imp__all__set,axiom,
    ! [Xs2: list_complex,P: complex > $o,X: complex] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
         => ( P @ ( nth_complex @ Xs2 @ I2 ) ) )
     => ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1176_all__nth__imp__all__set,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
         => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I2 ) ) )
     => ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1177_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1178_all__nth__imp__all__set,axiom,
    ! [Xs2: list_o,P: $o > $o,X: $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
         => ( P @ ( nth_o @ Xs2 @ I2 ) ) )
     => ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1179_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P: nat > $o,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I2 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1180_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P: int > $o,X: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I2 ) ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1181_in__set__conv__nth,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1182_in__set__conv__nth,axiom,
    ! [X: complex,Xs2: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
            & ( ( nth_complex @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1183_in__set__conv__nth,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
            & ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1184_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1185_in__set__conv__nth,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1186_in__set__conv__nth,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1187_in__set__conv__nth,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I3 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1188_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ! [X5: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_1189_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_1190_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_1191_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_int,P: int > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_1192_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_1193_mul__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% mul_shift
thf(fact_1194_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_1195_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_1196_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_1197_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_1198_Suc__double__not__eq__double,axiom,
    ! [M2: nat,N: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_double_not_eq_double
thf(fact_1199_double__not__eq__Suc__double,axiom,
    ! [M2: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% double_not_eq_Suc_double
thf(fact_1200_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_1201_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_1202_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_1203_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_1204_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_1205_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_1206_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_1207_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_1208_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_1209_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_1210_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_1211_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_1212_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_1213_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X ) ) ).

% min_Null_member
thf(fact_1214_semiring__norm_I13_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M2 @ N ) ) ) ) ).

% semiring_norm(13)
thf(fact_1215_semiring__norm_I11_J,axiom,
    ! [M2: num] :
      ( ( times_times_num @ M2 @ one )
      = M2 ) ).

% semiring_norm(11)
thf(fact_1216_semiring__norm_I12_J,axiom,
    ! [N: num] :
      ( ( times_times_num @ one @ N )
      = N ) ).

% semiring_norm(12)
thf(fact_1217_num__double,axiom,
    ! [N: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N )
      = ( bit0 @ N ) ) ).

% num_double
thf(fact_1218_power__mult__numeral,axiom,
    ! [A: nat,M2: num,N: num] :
      ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_1219_power__mult__numeral,axiom,
    ! [A: real,M2: num,N: num] :
      ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_1220_power__mult__numeral,axiom,
    ! [A: int,M2: num,N: num] :
      ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_1221_power__mult__numeral,axiom,
    ! [A: complex,M2: num,N: num] :
      ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_1222_four__x__squared,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% four_x_squared
thf(fact_1223_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_1224_div__mult2__numeral__eq,axiom,
    ! [A: nat,K2: num,L: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K2 ) ) @ ( numeral_numeral_nat @ L ) )
      = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K2 @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_1225_div__mult2__numeral__eq,axiom,
    ! [A: int,K2: num,L: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ L ) )
      = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K2 @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_1226_two__realpow__ge__one,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% two_realpow_ge_one
thf(fact_1227_linordered__field__no__ub,axiom,
    ! [X3: real] :
    ? [X_12: real] : ( ord_less_real @ X3 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_1228_linordered__field__no__ub,axiom,
    ! [X3: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X3 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_1229_linordered__field__no__lb,axiom,
    ! [X3: real] :
    ? [Y4: real] : ( ord_less_real @ Y4 @ X3 ) ).

% linordered_field_no_lb
thf(fact_1230_linordered__field__no__lb,axiom,
    ! [X3: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X3 ) ).

% linordered_field_no_lb
thf(fact_1231_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
      = none_nat ) ).

% vebt_succ.simps(3)
thf(fact_1232_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_1233_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_1234_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_1235_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_1236_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_1237_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1238_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1239_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1240_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1241_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] :
      ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1242_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1243_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_1244_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_1245_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_1246_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
      = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1247_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_1248_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1249_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2 = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M2 ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_1250_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M2 ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_1251_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_1252_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_1253_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_1254_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).

% gt_half_sum
thf(fact_1255_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_1256_real__average__minus__first,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_first
thf(fact_1257_real__average__minus__second,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_second
thf(fact_1258_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_1259_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_1260_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q3: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_1261_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q3: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_1262_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q3: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_1263_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_1264_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_1265_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_1266_discrete,axiom,
    ( ord_less_int
    = ( ^ [A5: int] : ( ord_less_eq_int @ ( plus_plus_int @ A5 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_1267_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X4: nat,N3: nat] : ( modulo_modulo_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% low_def
thf(fact_1268_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_1269_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit0
thf(fact_1270_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_1271_mod__mod__trivial,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_1272_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_1273_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_1274_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_1275_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_1276_minus__mod__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mod_self2
thf(fact_1277_real__divide__square__eq,axiom,
    ! [R2: real,A: real] :
      ( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
      = ( divide_divide_real @ A @ R2 ) ) ).

% real_divide_square_eq
thf(fact_1278_mod__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ M2 @ N )
        = M2 ) ) ).

% mod_less
thf(fact_1279_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_1280_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_1281_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_1282_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_1283_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_1284_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_1285_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_1286_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_1287_Suc__mod__mult__self4,axiom,
    ! [N: nat,K2: nat,M2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K2 ) @ M2 ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_mod_mult_self4
thf(fact_1288_Suc__mod__mult__self3,axiom,
    ! [K2: nat,N: nat,M2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K2 @ N ) @ M2 ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_mod_mult_self3
thf(fact_1289_Suc__mod__mult__self2,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ N @ K2 ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_mod_mult_self2
thf(fact_1290_Suc__mod__mult__self1,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ K2 @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_mod_mult_self1
thf(fact_1291_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_mod_two_eq_one
thf(fact_1292_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_mod_two_eq_one
thf(fact_1293_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% bits_one_mod_two_eq_one
thf(fact_1294_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_one_mod_two_eq_one
thf(fact_1295_mod2__Suc__Suc,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% mod2_Suc_Suc
thf(fact_1296_Suc__times__numeral__mod__eq,axiom,
    ! [K2: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K2 )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K2 ) @ N ) ) @ ( numeral_numeral_nat @ K2 ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_1297_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).

% real_arch_pow
thf(fact_1298_mult__commute__abs,axiom,
    ! [C: real] :
      ( ( ^ [X4: real] : ( times_times_real @ X4 @ C ) )
      = ( times_times_real @ C ) ) ).

% mult_commute_abs
thf(fact_1299_mult__commute__abs,axiom,
    ! [C: rat] :
      ( ( ^ [X4: rat] : ( times_times_rat @ X4 @ C ) )
      = ( times_times_rat @ C ) ) ).

% mult_commute_abs
thf(fact_1300_mult__commute__abs,axiom,
    ! [C: nat] :
      ( ( ^ [X4: nat] : ( times_times_nat @ X4 @ C ) )
      = ( times_times_nat @ C ) ) ).

% mult_commute_abs
thf(fact_1301_mult__commute__abs,axiom,
    ! [C: int] :
      ( ( ^ [X4: int] : ( times_times_int @ X4 @ C ) )
      = ( times_times_int @ C ) ) ).

% mult_commute_abs
thf(fact_1302_lambda__one,axiom,
    ( ( ^ [X4: complex] : X4 )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_1303_lambda__one,axiom,
    ( ( ^ [X4: real] : X4 )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_1304_lambda__one,axiom,
    ( ( ^ [X4: rat] : X4 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_1305_lambda__one,axiom,
    ( ( ^ [X4: nat] : X4 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_1306_lambda__one,axiom,
    ( ( ^ [X4: int] : X4 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_1307_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_1308_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_1309_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_1310_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_1311_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_1312_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] :
      ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
      = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_1313_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_1314_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_1315_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A2 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B2 @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1316_mod__mult__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A2 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B2 @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1317_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_1318_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_1319_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_1320_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_1321_mod__add__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A2 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B2 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1322_mod__add__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A2 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B2 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1323_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_1324_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_1325_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_1326_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_1327_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_1328_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_1329_mod__diff__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A2 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B2 @ C ) )
       => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( minus_minus_int @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_1330_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_eq
thf(fact_1331_power__mod,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
      = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_1332_power__mod,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
      = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_1333_mod__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).

% mod_Suc_eq
thf(fact_1334_mod__Suc__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ N ) ) ).

% mod_Suc_Suc_eq
thf(fact_1335_mod__less__eq__dividend,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ M2 ) ).

% mod_less_eq_dividend
thf(fact_1336_cong__exp__iff__simps_I9_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ Q3 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1337_cong__exp__iff__simps_I9_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ Q3 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1338_cong__exp__iff__simps_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ one ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1339_cong__exp__iff__simps_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ one ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1340_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_code(2)
thf(fact_1341_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_code(2)
thf(fact_1342_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_code(2)
thf(fact_1343_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_code(2)
thf(fact_1344_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_code(2)
thf(fact_1345_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_1346_mod__geq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ ( ord_less_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ M2 @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ).

% mod_geq
thf(fact_1347_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_1348_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_1349_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_1350_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_1351_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_1352_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_1353_cong__exp__iff__simps_I8_J,axiom,
    ! [M2: num,Q3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1354_cong__exp__iff__simps_I8_J,axiom,
    ! [M2: num,Q3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1355_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1356_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1357_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D3: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_1358_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_1359_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_1360_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P2: nat,M2: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P2 )
       => ( ( ord_less_nat @ M2 @ P2 )
         => ( ! [N2: nat] :
                ( ( ord_less_nat @ N2 @ P2 )
               => ( ( P @ N2 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P2 ) ) ) )
           => ( P @ M2 ) ) ) ) ) ).

% mod_induct
thf(fact_1361_mod__Suc__le__divisor,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ ( suc @ N ) ) @ N ) ).

% mod_Suc_le_divisor
thf(fact_1362_mod__eq__nat1E,axiom,
    ! [M2: nat,Q3: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ Q3 )
        = ( modulo_modulo_nat @ N @ Q3 ) )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ~ ! [S: nat] :
              ( M2
             != ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_1363_mod__eq__nat2E,axiom,
    ! [M2: nat,Q3: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ Q3 )
        = ( modulo_modulo_nat @ N @ Q3 ) )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ~ ! [S: nat] :
              ( N
             != ( plus_plus_nat @ M2 @ ( times_times_nat @ Q3 @ S ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_1364_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q2: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q2 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_1365_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N3 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ).

% mod_if
thf(fact_1366_le__mod__geq,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( modulo_modulo_nat @ M2 @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ).

% le_mod_geq
thf(fact_1367_set__conv__nth,axiom,
    ( set_complex2
    = ( ^ [Xs: list_complex] :
          ( collect_complex
          @ ^ [Uu3: complex] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_complex @ Xs @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1368_set__conv__nth,axiom,
    ( set_real2
    = ( ^ [Xs: list_real] :
          ( collect_real
          @ ^ [Uu3: real] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_real @ Xs @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1369_set__conv__nth,axiom,
    ( set_list_nat2
    = ( ^ [Xs: list_list_nat] :
          ( collect_list_nat
          @ ^ [Uu3: list_nat] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_list_nat @ Xs @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1370_set__conv__nth,axiom,
    ( set_set_nat2
    = ( ^ [Xs: list_set_nat] :
          ( collect_set_nat
          @ ^ [Uu3: set_nat] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_set_nat @ Xs @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1371_set__conv__nth,axiom,
    ( set_VEBT_VEBT2
    = ( ^ [Xs: list_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [Uu3: vEBT_VEBT] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_VEBT_VEBT @ Xs @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1372_set__conv__nth,axiom,
    ( set_o2
    = ( ^ [Xs: list_o] :
          ( collect_o
          @ ^ [Uu3: $o] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_o @ Xs @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1373_set__conv__nth,axiom,
    ( set_nat2
    = ( ^ [Xs: list_nat] :
          ( collect_nat
          @ ^ [Uu3: nat] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_nat @ Xs @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1374_set__conv__nth,axiom,
    ( set_int2
    = ( ^ [Xs: list_int] :
          ( collect_int
          @ ^ [Uu3: int] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_int @ Xs @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1375_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_1376_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_1377_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_1378_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_1379_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_1380_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_1381_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_1382_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_1383_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_1384_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_1385_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_1386_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_1387_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_1388_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_1389_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_1390_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_1391_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_1392_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_1393_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_1394_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_1395_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_1396_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_1397_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_1398_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_1399_mod__mult2__eq,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( times_times_nat @ N @ Q3 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M2 @ N ) @ Q3 ) ) @ ( modulo_modulo_nat @ M2 @ N ) ) ) ).

% mod_mult2_eq
thf(fact_1400_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M3: nat,N3: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N3 ) @ N3 ) ) ) ) ).

% modulo_nat_def
thf(fact_1401_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: nat,M2: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1402_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M2: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1403_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_1404_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_1405_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( 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 ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_1406_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: 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 ) ) @ TreeList @ 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 @ TreeList ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_1407_buildup__gives__empty,axiom,
    ! [N: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_1408_Diff__eq__empty__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ( minus_minus_set_int @ A4 @ B4 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_1409_Diff__eq__empty__iff,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ( ( minus_minus_set_real @ A4 @ B4 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_1410_Diff__eq__empty__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ( minus_minus_set_nat @ A4 @ B4 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_1411_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ 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 @ TreeList ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_1412_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S2 ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_1413_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_1414_div__mod__decomp,axiom,
    ! [A4: nat,N: nat] :
      ( A4
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A4 @ N ) @ N ) @ ( modulo_modulo_nat @ A4 @ N ) ) ) ).

% div_mod_decomp
thf(fact_1415_set__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_1416_set__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_1417_flip__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_1418_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_1419_unset__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_1420_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_1421_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).

% buildup_nothing_in_min_max
thf(fact_1422_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).

% buildup_nothing_in_leaf
thf(fact_1423_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X4: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X4 )
          | ( vEBT_VEBT_membermima @ T2 @ X4 ) ) ) ) ).

% both_member_options_def
thf(fact_1424_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( bit0 @ X2 )
        = ( bit0 @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
thf(fact_1425_Diff__idemp,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ B4 )
      = ( minus_minus_set_nat @ A4 @ B4 ) ) ).

% Diff_idemp
thf(fact_1426_Diff__iff,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) )
      = ( ( member_int @ C @ A4 )
        & ~ ( member_int @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1427_Diff__iff,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) )
      = ( ( member_real @ C @ A4 )
        & ~ ( member_real @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1428_Diff__iff,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
      = ( ( member_complex @ C @ A4 )
        & ~ ( member_complex @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1429_Diff__iff,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
      = ( ( member8440522571783428010at_nat @ C @ A4 )
        & ~ ( member8440522571783428010at_nat @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1430_Diff__iff,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
      = ( ( member_nat @ C @ A4 )
        & ~ ( member_nat @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1431_DiffI,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ A4 )
     => ( ~ ( member_int @ C @ B4 )
       => ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1432_DiffI,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ A4 )
     => ( ~ ( member_real @ C @ B4 )
       => ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1433_DiffI,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ A4 )
     => ( ~ ( member_complex @ C @ B4 )
       => ( member_complex @ C @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1434_DiffI,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ A4 )
     => ( ~ ( member8440522571783428010at_nat @ C @ B4 )
       => ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1435_DiffI,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ A4 )
     => ( ~ ( member_nat @ C @ B4 )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1436_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N )
     => ( ( vEBT_vebt_member @ Tree @ X )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X )
          | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).

% member_valid_both_member_options
thf(fact_1437_Diff__empty,axiom,
    ! [A4: set_int] :
      ( ( minus_minus_set_int @ A4 @ bot_bot_set_int )
      = A4 ) ).

% Diff_empty
thf(fact_1438_Diff__empty,axiom,
    ! [A4: set_real] :
      ( ( minus_minus_set_real @ A4 @ bot_bot_set_real )
      = A4 ) ).

% Diff_empty
thf(fact_1439_Diff__empty,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ bot_bot_set_nat )
      = A4 ) ).

% Diff_empty
thf(fact_1440_empty__Diff,axiom,
    ! [A4: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A4 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_1441_empty__Diff,axiom,
    ! [A4: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A4 )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_1442_empty__Diff,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A4 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_1443_Diff__cancel,axiom,
    ! [A4: set_int] :
      ( ( minus_minus_set_int @ A4 @ A4 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_1444_Diff__cancel,axiom,
    ! [A4: set_real] :
      ( ( minus_minus_set_real @ A4 @ A4 )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_1445_Diff__cancel,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ A4 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_1446_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).

% zmod_numeral_Bit0
thf(fact_1447_psubset__imp__ex__mem,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_set_int @ A4 @ B4 )
     => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1448_psubset__imp__ex__mem,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ( ord_less_set_real @ A4 @ B4 )
     => ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1449_psubset__imp__ex__mem,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ( ord_less_set_complex @ A4 @ B4 )
     => ? [B3: complex] : ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1450_psubset__imp__ex__mem,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A4 @ B4 )
     => ? [B3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1451_psubset__imp__ex__mem,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B4 )
     => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1452_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X4: int] : ( member_int @ X4 @ A6 )
            @ ^ [X4: int] : ( member_int @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1453_minus__set__def,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( minus_2270307095948843157_nat_o
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A6 )
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1454_minus__set__def,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( collect_complex
          @ ( minus_8727706125548526216plex_o
            @ ^ [X4: complex] : ( member_complex @ X4 @ A6 )
            @ ^ [X4: complex] : ( member_complex @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1455_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X4: real] : ( member_real @ X4 @ A6 )
            @ ^ [X4: real] : ( member_real @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1456_minus__set__def,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ( minus_1139252259498527702_nat_o
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A6 )
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1457_minus__set__def,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ( minus_6910147592129066416_nat_o
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A6 )
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1458_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A6 )
            @ ^ [X4: nat] : ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1459_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A6 )
              & ~ ( member_int @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1460_set__diff__eq,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A6 )
              & ~ ( member8440522571783428010at_nat @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1461_set__diff__eq,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( collect_complex
          @ ^ [X4: complex] :
              ( ( member_complex @ X4 @ A6 )
              & ~ ( member_complex @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1462_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ^ [X4: real] :
              ( ( member_real @ X4 @ A6 )
              & ~ ( member_real @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1463_set__diff__eq,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A6: set_list_nat,B6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( member_list_nat @ X4 @ A6 )
              & ~ ( member_list_nat @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1464_set__diff__eq,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( member_set_nat @ X4 @ A6 )
              & ~ ( member_set_nat @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1465_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A6 )
              & ~ ( member_nat @ X4 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1466_DiffD2,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) )
     => ~ ( member_int @ C @ B4 ) ) ).

% DiffD2
thf(fact_1467_DiffD2,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) )
     => ~ ( member_real @ C @ B4 ) ) ).

% DiffD2
thf(fact_1468_DiffD2,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
     => ~ ( member_complex @ C @ B4 ) ) ).

% DiffD2
thf(fact_1469_DiffD2,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
     => ~ ( member8440522571783428010at_nat @ C @ B4 ) ) ).

% DiffD2
thf(fact_1470_DiffD2,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
     => ~ ( member_nat @ C @ B4 ) ) ).

% DiffD2
thf(fact_1471_DiffD1,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) )
     => ( member_int @ C @ A4 ) ) ).

% DiffD1
thf(fact_1472_DiffD1,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) )
     => ( member_real @ C @ A4 ) ) ).

% DiffD1
thf(fact_1473_DiffD1,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
     => ( member_complex @ C @ A4 ) ) ).

% DiffD1
thf(fact_1474_DiffD1,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
     => ( member8440522571783428010at_nat @ C @ A4 ) ) ).

% DiffD1
thf(fact_1475_DiffD1,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
     => ( member_nat @ C @ A4 ) ) ).

% DiffD1
thf(fact_1476_DiffE,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) )
     => ~ ( ( member_int @ C @ A4 )
         => ( member_int @ C @ B4 ) ) ) ).

% DiffE
thf(fact_1477_DiffE,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) )
     => ~ ( ( member_real @ C @ A4 )
         => ( member_real @ C @ B4 ) ) ) ).

% DiffE
thf(fact_1478_DiffE,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
     => ~ ( ( member_complex @ C @ A4 )
         => ( member_complex @ C @ B4 ) ) ) ).

% DiffE
thf(fact_1479_DiffE,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
     => ~ ( ( member8440522571783428010at_nat @ C @ A4 )
         => ( member8440522571783428010at_nat @ C @ B4 ) ) ) ).

% DiffE
thf(fact_1480_DiffE,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
     => ~ ( ( member_nat @ C @ A4 )
         => ( member_nat @ C @ B4 ) ) ) ).

% DiffE
thf(fact_1481_div__mod__decomp__int,axiom,
    ! [A4: int,N: int] :
      ( A4
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A4 @ N ) @ N ) @ ( modulo_modulo_int @ A4 @ N ) ) ) ).

% div_mod_decomp_int
thf(fact_1482_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ ( ord_less_eq_rat @ A @ B )
      | ~ ( ord_less_eq_rat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1483_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ ( ord_less_eq_num @ A @ B )
      | ~ ( ord_less_eq_num @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1484_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_1485_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_1486_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1487_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1488_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1489_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1490_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1491_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1492_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1493_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1494_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1495_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1496_double__diff,axiom,
    ! [A4: set_nat,B4: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ C4 )
       => ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C4 @ A4 ) )
          = A4 ) ) ) ).

% double_diff
thf(fact_1497_Diff__subset,axiom,
    ! [A4: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ A4 ) ).

% Diff_subset
thf(fact_1498_Diff__mono,axiom,
    ! [A4: set_nat,C4: set_nat,D4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ C4 )
     => ( ( ord_less_eq_set_nat @ D4 @ B4 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( minus_minus_set_nat @ C4 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_1499_verit__comp__simplify1_I3_J,axiom,
    ! [B2: real,A2: real] :
      ( ( ~ ( ord_less_eq_real @ B2 @ A2 ) )
      = ( ord_less_real @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1500_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ~ ( ord_less_eq_rat @ B2 @ A2 ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1501_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ( ~ ( ord_less_eq_num @ B2 @ A2 ) )
      = ( ord_less_num @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1502_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1503_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ( ~ ( ord_less_eq_int @ B2 @ A2 ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1504_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] :
      ( one
     != ( bit0 @ X2 ) ) ).

% verit_eq_simplify(10)
thf(fact_1505_vebt__member_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X ) ).

% vebt_member.simps(2)
thf(fact_1506_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_1507_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_1508_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ 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 @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_1509_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ! [Uu: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ Uu @ B3 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( B3
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ Uv @ Uw ) )
           => ( ? [N2: nat] :
                  ( Xa2
                  = ( suc @ N2 ) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                      = none_nat )
                                    @ none_nat
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_1510_product__nth,axiom,
    ! [N: nat,Xs2: list_num,Ys: list_num] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_num @ Xs2 ) @ ( size_size_list_num @ Ys ) ) )
     => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs2 @ Ys ) @ N )
        = ( product_Pair_num_num @ ( nth_num @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_1511_product__nth,axiom,
    ! [N: nat,Xs2: list_Code_integer,Ys: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr8522763379788166057eger_o @ ( produc3607205314601156340eger_o @ Xs2 @ Ys ) @ N )
        = ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_1512_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) @ N )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_1513_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) @ N )
        = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_1514_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) @ N )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_1515_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) @ N )
        = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_1516_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) @ N )
        = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_1517_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys ) @ N )
        = ( product_Pair_o_o @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_1518_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys ) @ N )
        = ( product_Pair_o_nat @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_1519_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys ) @ N )
        = ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_1520_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_1521_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A4 @ Z )
       => ( ( finite_finite_nat @ B4 )
         => ( ( A4 = B4 )
           => ? [X_12: nat] : ( vEBT_is_succ_in_set @ A4 @ X @ X_12 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_1522_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).

% buildup_gives_valid
thf(fact_1523_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList: 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 @ TreeList ) )
     => ( ( 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 @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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_1524_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: 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 @ TreeList ) )
     => ( ( 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 @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_1525_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_1526_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_1527_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_1528_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_1529_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% deg_not_0
thf(fact_1530_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_1531_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A5: $o,B5: $o] :
            ( T
            = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ).

% deg1Leaf
thf(fact_1532_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A3: $o,B3: $o] :
          ( T
          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ).

% deg_1_Leaf
thf(fact_1533_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( N = one_one_nat )
       => ? [A3: $o,B3: $o] :
            ( T
            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).

% deg_1_Leafy
thf(fact_1534_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_1535_succ__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_12: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_12 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X3: nat] :
              ( ( member_nat @ X3 @ Xs2 )
              & ( ord_less_nat @ A @ X3 ) ) ) ) ).

% succ_none_empty
thf(fact_1536_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
      ( ( ( vEBT_Leaf @ X21 @ X222 )
        = ( vEBT_Leaf @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% VEBT.inject(2)
thf(fact_1537_list__update__overwrite,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ I @ Y )
      = ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ Y ) ) ).

% list_update_overwrite
thf(fact_1538_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1539_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_1540_mult__zero__left,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_1541_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_1542_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_1543_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_1544_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_1545_mult__zero__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_1546_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_1547_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_1548_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_1549_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_1550_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_1551_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_1552_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_1553_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_1554_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_1555_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_1556_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_1557_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( times_times_rat @ C @ A )
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_1558_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_1559_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_1560_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_1561_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_1562_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( times_times_rat @ A @ C )
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_1563_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_1564_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_1565_add_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.right_neutral
thf(fact_1566_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_1567_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_1568_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_1569_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_1570_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_1571_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_1572_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_1573_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_1574_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_1575_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_1576_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_1577_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_1578_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_1579_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_1580_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_1581_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_1582_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_1583_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_1584_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_1585_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ B @ A ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_1586_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_1587_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_1588_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_1589_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_1590_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ A @ B ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_1591_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_1592_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_1593_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_1594_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_1595_add__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add_0
thf(fact_1596_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_1597_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_1598_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_1599_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_1600_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_1601_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_1602_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1603_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_1604_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_1605_diff__zero,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_zero
thf(fact_1606_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_1607_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_1608_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_1609_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_1610_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1611_diff__0__right,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_0_right
thf(fact_1612_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_1613_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_1614_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_1615_diff__self,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% diff_self
thf(fact_1616_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_1617_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_1618_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_1619_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_1620_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_1621_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_1622_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_1623_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_1624_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_1625_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_1626_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_1627_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_1628_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_1629_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_1630_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_1631_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_1632_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_1633_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_1634_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_1635_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
thf(fact_1636_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_1637_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_1638_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( divide_divide_rat @ C @ A )
        = ( divide_divide_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_1639_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_1640_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_1641_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_1642_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_1643_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_1644_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_1645_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_1646_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_1647_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_1648_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_1649_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_1650_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_1651_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_1652_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_1653_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_1654_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_1655_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1656_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1657_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1658_List_Ofinite__set,axiom,
    ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_1659_List_Ofinite__set,axiom,
    ! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_1660_List_Ofinite__set,axiom,
    ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_1661_List_Ofinite__set,axiom,
    ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_1662_add__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_1663_Nat_Oadd__0__right,axiom,
    ! [M2: nat] :
      ( ( plus_plus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% Nat.add_0_right
thf(fact_1664_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1665_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1666_mult__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ K2 )
        = ( times_times_nat @ N @ K2 ) )
      = ( ( M2 = N )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_1667_mult__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ( times_times_nat @ K2 @ M2 )
        = ( times_times_nat @ K2 @ N ) )
      = ( ( M2 = N )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_1668_mult__0__right,axiom,
    ! [M2: nat] :
      ( ( times_times_nat @ M2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_1669_mult__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_1670_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_list_update
thf(fact_1671_length__list__update,axiom,
    ! [Xs2: list_o,I: nat,X: $o] :
      ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I @ X ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_list_update
thf(fact_1672_length__list__update,axiom,
    ! [Xs2: list_nat,I: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I @ X ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_list_update
thf(fact_1673_length__list__update,axiom,
    ! [Xs2: list_int,I: nat,X: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs2 @ I @ X ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_list_update
thf(fact_1674_max__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_max_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ M2 @ N ) ) ) ).

% max_Suc_Suc
thf(fact_1675_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_1676_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_1677_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_1678_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_1679_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_1680_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_1681_list__update__id,axiom,
    ! [Xs2: list_nat,I: nat] :
      ( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_1682_list__update__id,axiom,
    ! [Xs2: list_int,I: nat] :
      ( ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_1683_list__update__id,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat] :
      ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_1684_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_nat,X: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
        = ( nth_nat @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_1685_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_int,X: int] :
      ( ( I != J )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ J )
        = ( nth_int @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_1686_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I != J )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
        = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_1687_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_1688_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_1689_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_1690_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_1691_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_1692_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_1693_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_1694_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_1695_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_1696_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_1697_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel1
thf(fact_1698_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_1699_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_1700_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_1701_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel2
thf(fact_1702_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_1703_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_1704_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_1705_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1706_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_1707_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_1708_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1709_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_1710_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_1711_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1712_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_1713_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_1714_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_1715_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_1716_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_1717_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1718_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_1719_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_1720_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel2
thf(fact_1721_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_1722_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_1723_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_1724_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel1
thf(fact_1725_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_1726_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_1727_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_1728_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_1729_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_1730_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_1731_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_1732_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_1733_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_1734_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_1735_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_1736_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1737_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_1738_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_1739_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_1740_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_1741_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_1742_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_1743_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_1744_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ( times_times_rat @ C @ A )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_1745_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_1746_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_1747_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_1748_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_1749_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_1750_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_1751_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_1752_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ( times_times_rat @ A @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_1753_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_1754_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_1755_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1756_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_1757_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_1758_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_1759_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1760_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1761_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_1762_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_1763_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_1764_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1765_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_1766_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_1767_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_1768_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_1769_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1770_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_1771_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_1772_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_1773_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_1774_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_1775_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_1776_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_1777_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_1778_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_1779_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_1780_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = ( divide_divide_rat @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1781_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_1782_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_1783_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1784_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_1785_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_1786_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1787_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_1788_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_1789_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1790_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_1791_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_1792_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1793_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_1794_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_1795_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_1796_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_1797_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_1798_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_1799_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_1800_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = one_one_rat )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_1801_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_1802_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_1803_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A @ B ) )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_1804_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_1805_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_1806_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_1807_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_1808_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_1809_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = one_one_rat ) ) ) ).

% divide_self_if
thf(fact_1810_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_1811_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ A )
        = one_one_rat )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_1812_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_1813_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B @ A ) )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_1814_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_1815_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( divide_divide_rat @ one_one_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
thf(fact_1816_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_1817_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( divide_divide_rat @ one_one_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
thf(fact_1818_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_1819_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_1820_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_1821_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_1822_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_1823_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_1824_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_1825_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_1826_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_1827_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_1828_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_1829_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_1830_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_1831_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_1832_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_1833_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_1834_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_1835_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_1836_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_1837_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_1838_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_1839_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_1840_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1841_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1842_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1843_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1844_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1845_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1846_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_1847_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_1848_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_1849_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_1850_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_1851_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_1852_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_1853_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(3)
thf(fact_1854_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_1855_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_1856_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_1857_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_1858_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(4)
thf(fact_1859_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_1860_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_1861_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_1862_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_1863_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_1864_max__0__1_I1_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(1)
thf(fact_1865_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_1866_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_1867_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_1868_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_1869_max__0__1_I2_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(2)
thf(fact_1870_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_1871_add__gr__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_1872_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_1873_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_1874_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(5)
thf(fact_1875_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_1876_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_1877_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_1878_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_1879_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(6)
thf(fact_1880_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_1881_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_1882_mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M2
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_1883_one__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M2 @ N ) )
      = ( ( M2
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_1884_zero__less__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% zero_less_diff
thf(fact_1885_mult__less__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
        & ( ord_less_nat @ M2 @ N ) ) ) ).

% mult_less_cancel2
thf(fact_1886_nat__0__less__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_1887_nat__mult__less__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
        & ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_1888_div__by__Suc__0,axiom,
    ! [M2: nat] :
      ( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = M2 ) ).

% div_by_Suc_0
thf(fact_1889_diff__is__0__eq_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1890_diff__is__0__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% diff_is_0_eq
thf(fact_1891_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1892_div__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( divide_divide_nat @ M2 @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_1893_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M2: nat] :
      ( ( ( power_power_nat @ X @ M2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M2 = zero_zero_nat )
        | ( X
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_1894_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_1895_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_1896_mod__by__Suc__0,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_1897_nat__mult__div__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ( K2 = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
          = zero_zero_nat ) )
      & ( ( K2 != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
          = ( divide_divide_nat @ M2 @ N ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_1898_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_1899_list__update__beyond,axiom,
    ! [Xs2: list_o,I: nat,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
     => ( ( list_update_o @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_1900_list__update__beyond,axiom,
    ! [Xs2: list_nat,I: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
     => ( ( list_update_nat @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_1901_list__update__beyond,axiom,
    ! [Xs2: list_int,I: nat,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I )
     => ( ( list_update_int @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_1902_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_1903_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_1904_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_nat] :
      ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_1905_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_int] :
      ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_1906_length__product,axiom,
    ! [Xs2: list_o,Ys: list_VEBT_VEBT] :
      ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_1907_length__product,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_1908_length__product,axiom,
    ! [Xs2: list_o,Ys: list_nat] :
      ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_1909_length__product,axiom,
    ! [Xs2: list_o,Ys: list_int] :
      ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_1910_length__product,axiom,
    ! [Xs2: list_nat,Ys: list_VEBT_VEBT] :
      ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_1911_length__product,axiom,
    ! [Xs2: list_nat,Ys: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_1912_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_1913_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_1914_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% divide_le_0_1_iff
thf(fact_1915_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_1916_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_1917_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_1918_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_1919_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_1920_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_1921_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_1922_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_1923_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_1924_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_1925_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_1926_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_1927_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_1928_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_1929_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
        = A )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1930_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1931_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1932_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
            = B ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1933_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
            = B ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1934_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
            = B ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1935_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_1936_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_1937_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_1938_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_1939_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_1940_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_1941_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_1942_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_1943_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_1944_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_1945_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_1946_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_1947_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_1948_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_1949_power__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ( power_power_rat @ A @ N )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1950_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( ( power_power_nat @ A @ N )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1951_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ( power_power_real @ A @ N )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1952_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( ( power_power_int @ A @ N )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1953_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( ( power_power_complex @ A @ N )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1954_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_1955_one__le__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_1956_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_1957_mult__le__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% mult_le_cancel2
thf(fact_1958_div__mult__self1__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
        = M2 ) ) ).

% div_mult_self1_is_m
thf(fact_1959_div__mult__self__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
        = M2 ) ) ).

% div_mult_self_is_m
thf(fact_1960_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1961_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1962_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1963_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1964_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_1965_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_1966_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_1967_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_1968_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_1969_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_1970_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_1971_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_1972_power__strict__decreasing__iff,axiom,
    ! [B: real,M2: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1973_power__strict__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M2 ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1974_power__strict__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1975_power__strict__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1976_power__mono__iff,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
            = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1977_power__mono__iff,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
            = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1978_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1979_power__mono__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
            = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1980_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_power2
thf(fact_1981_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_1982_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_1983_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_1984_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_1985_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_1986_set__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) )
          = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_1987_set__swap,axiom,
    ! [I: nat,Xs2: list_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I ) ) )
          = ( set_o2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_1988_set__swap,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
          = ( set_nat2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_1989_set__swap,axiom,
    ! [I: nat,Xs2: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I ) ) )
          = ( set_int2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_1990_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_1991_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_1992_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_1993_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_1994_power__decreasing__iff,axiom,
    ! [B: real,M2: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_1995_power__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M2 ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_1996_power__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_1997_power__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_1998_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_1999_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
thf(fact_2000_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_2001_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_2002_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2003_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_2004_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_2005_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_2006_zero__less__power2,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_power2
thf(fact_2007_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_2008_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2009_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_2010_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_2011_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_2012_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_2013_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_2014_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_2015_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != ( suc @ zero_zero_nat ) )
      = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_2016_add__self__mod__2,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% add_self_mod_2
thf(fact_2017_mod2__gr__0,axiom,
    ! [M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% mod2_gr_0
thf(fact_2018_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_2019_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_2020_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
      = ( X = zero_zero_complex ) ) ).

% zero_reorient
thf(fact_2021_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_2022_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_2023_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_2024_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_2025_list__update__swap,axiom,
    ! [I: nat,I5: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT,X8: vEBT_VEBT] :
      ( ( I != I5 )
     => ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ I5 @ X8 )
        = ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I5 @ X8 ) @ I @ X ) ) ) ).

% list_update_swap
thf(fact_2026_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_2027_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_2028_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
     => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X5: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ X5 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_2029_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_2030_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_2031_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_2032_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_2033_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_2034_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_2035_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_2036_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_2037_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_2038_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_2039_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_2040_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_2041_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_2042_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_2043_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_2044_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_2045_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_2046_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_rat @ zero_zero_rat @ N )
        = zero_zero_rat ) ) ).

% zero_power
thf(fact_2047_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_nat @ zero_zero_nat @ N )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_2048_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_real @ zero_zero_real @ N )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_2049_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_int @ zero_zero_int @ N )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_2050_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_complex @ zero_zero_complex @ N )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_2051_nat__add__max__left,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M2 @ N ) @ Q3 )
      = ( ord_max_nat @ ( plus_plus_nat @ M2 @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).

% nat_add_max_left
thf(fact_2052_nat__add__max__right,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( plus_plus_nat @ M2 @ ( ord_max_nat @ N @ Q3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M2 @ N ) @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ).

% nat_add_max_right
thf(fact_2053_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_2054_nat__mult__max__right,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ M2 @ ( ord_max_nat @ N @ Q3 ) )
      = ( ord_max_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q3 ) ) ) ).

% nat_mult_max_right
thf(fact_2055_nat__mult__max__left,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M2 @ N ) @ Q3 )
      = ( ord_max_nat @ ( times_times_nat @ M2 @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).

% nat_mult_max_left
thf(fact_2056_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o,D3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ D3 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_2057_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_2058_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_2059_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_2060_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_2061_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_2062_finite__list,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_2063_finite__list,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_2064_finite__list,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_2065_finite__list,axiom,
    ! [A4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_2066_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_2067_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_2068_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_2069_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_2070_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_2071_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_2072_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_2073_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_2074_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_2075_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_2076_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_2077_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_2078_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_2079_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D22 )
       => ? [E2: real] :
            ( ( ord_less_real @ zero_zero_real @ E2 )
            & ( ord_less_real @ E2 @ D1 )
            & ( ord_less_real @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2080_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D22 )
       => ? [E2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E2 )
            & ( ord_less_rat @ E2 @ D1 )
            & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2081_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N ) ) ).

% zero_neq_numeral
thf(fact_2082_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N ) ) ).

% zero_neq_numeral
thf(fact_2083_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N ) ) ).

% zero_neq_numeral
thf(fact_2084_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N ) ) ).

% zero_neq_numeral
thf(fact_2085_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N ) ) ).

% zero_neq_numeral
thf(fact_2086_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_2087_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_2088_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
       != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_2089_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_2090_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_2091_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_2092_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_2093_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_2094_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_2095_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_2096_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_2097_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_2098_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( times_times_rat @ A @ B )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_2099_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_2100_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_2101_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_2102_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_2103_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A )
          = ( times_times_rat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2104_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_2105_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_2106_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_2107_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_2108_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = ( times_times_rat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2109_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_2110_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_2111_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_2112_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_2113_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_2114_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_2115_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_2116_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_2117_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_2118_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2119_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_2120_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_2121_add_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.comm_neutral
thf(fact_2122_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_2123_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_2124_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_2125_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_2126_add_Ogroup__left__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2127_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2128_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2129_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2130_verit__sum__simplify,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% verit_sum_simplify
thf(fact_2131_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_2132_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_2133_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_2134_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_2135_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: complex,Z4: complex] : Y6 = Z4 )
    = ( ^ [A5: complex,B5: complex] :
          ( ( minus_minus_complex @ A5 @ B5 )
          = zero_zero_complex ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2136_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: real,Z4: real] : Y6 = Z4 )
    = ( ^ [A5: real,B5: real] :
          ( ( minus_minus_real @ A5 @ B5 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2137_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
    = ( ^ [A5: rat,B5: rat] :
          ( ( minus_minus_rat @ A5 @ B5 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2138_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [A5: int,B5: int] :
          ( ( minus_minus_int @ A5 @ B5 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2139_power__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( ( power_power_rat @ A @ N )
       != zero_zero_rat ) ) ).

% power_not_zero
thf(fact_2140_power__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N )
       != zero_zero_nat ) ) ).

% power_not_zero
thf(fact_2141_power__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N )
       != zero_zero_real ) ) ).

% power_not_zero
thf(fact_2142_power__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N )
       != zero_zero_int ) ) ).

% power_not_zero
thf(fact_2143_power__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( ( power_power_complex @ A @ N )
       != zero_zero_complex ) ) ).

% power_not_zero
thf(fact_2144_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_2145_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_2146_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_2147_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_2148_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_2149_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_2150_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_2151_diff__induct,axiom,
    ! [P: nat > nat > $o,M2: nat,N: nat] :
      ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
     => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
       => ( ! [X5: nat,Y4: nat] :
              ( ( P @ X5 @ Y4 )
             => ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
         => ( P @ M2 @ N ) ) ) ) ).

% diff_induct
thf(fact_2152_zero__induct,axiom,
    ! [P: nat > $o,K2: nat] :
      ( ( P @ K2 )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_2153_Suc__neq__Zero,axiom,
    ! [M2: nat] :
      ( ( suc @ M2 )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_2154_Zero__neq__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_neq_Suc
thf(fact_2155_Zero__not__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_not_Suc
thf(fact_2156_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M: nat] :
          ( N
          = ( suc @ M ) ) ) ).

% not0_implies_Suc
thf(fact_2157_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va3: nat] :
              ( X
             != ( suc @ ( suc @ Va3 ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_2158_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N2: nat] :
            ( X
           != ( suc @ N2 ) ) ) ).

% list_decode.cases
thf(fact_2159_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,Uw2: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) ).

% VEBT_internal.membermima.simps(1)
thf(fact_2160_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_2161_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_2162_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_2163_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_2164_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_2165_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_2166_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M4: nat] :
                  ( ( ord_less_nat @ M4 @ N2 )
                  & ~ ( P @ M4 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_2167_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_2168_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_2169_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_2170_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_2171_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_2172_add__eq__self__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = M2 )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_2173_diffs0__imp__equal,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M2 )
          = zero_zero_nat )
       => ( M2 = N ) ) ) ).

% diffs0_imp_equal
thf(fact_2174_minus__nat_Odiff__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% minus_nat.diff_0
thf(fact_2175_nat__mult__eq__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ( times_times_nat @ K2 @ M2 )
        = ( times_times_nat @ K2 @ N ) )
      = ( ( K2 = zero_zero_nat )
        | ( M2 = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_2176_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_2177_max__def__raw,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A5: extended_enat,B5: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).

% max_def_raw
thf(fact_2178_max__def__raw,axiom,
    ( ord_max_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).

% max_def_raw
thf(fact_2179_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A5: rat,B5: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).

% max_def_raw
thf(fact_2180_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A5: num,B5: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).

% max_def_raw
thf(fact_2181_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A5: nat,B5: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).

% max_def_raw
thf(fact_2182_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A5: int,B5: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).

% max_def_raw
thf(fact_2183_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ( power_power_real @ A @ N )
        = ( power_power_real @ B @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2184_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ( power_power_rat @ A @ N )
        = ( power_power_rat @ B @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2185_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N )
        = ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2186_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ( power_power_int @ A @ N )
        = ( power_power_int @ B @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2187_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ( power_power_real @ A @ N )
              = ( power_power_real @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2188_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ( power_power_rat @ A @ N )
              = ( power_power_rat @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2189_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N )
              = ( power_power_nat @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2190_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ( power_power_int @ A @ N )
              = ( power_power_int @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2191_lambda__zero,axiom,
    ( ( ^ [H: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_2192_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_2193_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_2194_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_2195_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_2196_finite__lists__length__eq,axiom,
    ! [A4: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 )
              & ( ( size_s3451745648224563538omplex @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_2197_finite__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_2198_finite__lists__length__eq,axiom,
    ! [A4: set_o,N: nat] :
      ( ( finite_finite_o @ A4 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 )
              & ( ( size_size_list_o @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_2199_finite__lists__length__eq,axiom,
    ! [A4: set_int,N: nat] :
      ( ( finite_finite_int @ A4 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
              & ( ( size_size_list_int @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_2200_finite__lists__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
              & ( ( size_size_list_nat @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_2201_nat__minus__add__max,axiom,
    ! [N: nat,M2: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M2 ) @ M2 )
      = ( ord_max_nat @ N @ M2 ) ) ).

% nat_minus_add_max
thf(fact_2202_power__strict__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_2203_power__strict__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_2204_power__strict__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_2205_power__strict__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_2206_set__update__subsetI,axiom,
    ! [Xs2: list_real,A4: set_real,X: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_2207_set__update__subsetI,axiom,
    ! [Xs2: list_complex,A4: set_complex,X: complex,I: nat] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A4 )
     => ( ( member_complex @ X @ A4 )
       => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_2208_set__update__subsetI,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I: nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A4 )
     => ( ( member8440522571783428010at_nat @ X @ A4 )
       => ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ I @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_2209_set__update__subsetI,axiom,
    ! [Xs2: list_int,A4: set_int,X: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_2210_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBT,A4: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A4 )
     => ( ( member_VEBT_VEBT @ X @ A4 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_2211_set__update__subsetI,axiom,
    ! [Xs2: list_nat,A4: set_nat,X: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_2212_finite__lists__length__le,axiom,
    ! [A4: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_2213_finite__lists__length__le,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_2214_finite__lists__length__le,axiom,
    ! [A4: set_o,N: nat] :
      ( ( finite_finite_o @ A4 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_2215_finite__lists__length__le,axiom,
    ! [A4: set_int,N: nat] :
      ( ( finite_finite_int @ A4 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_2216_finite__lists__length__le,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_2217_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_le_numeral
thf(fact_2218_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_le_numeral
thf(fact_2219_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_le_numeral
thf(fact_2220_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_le_numeral
thf(fact_2221_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_2222_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_2223_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_2224_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_2225_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_2226_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2227_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_2228_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_2229_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_2230_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2231_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_2232_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_2233_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2234_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_2235_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_2236_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_2237_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2238_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_2239_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_2240_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_2241_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2242_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_2243_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_2244_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_2245_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2246_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_2247_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_2248_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_2249_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_2250_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_2251_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_2252_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_2253_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2254_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_2255_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_2256_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2257_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_2258_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_2259_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_2260_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2261_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_2262_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_2263_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2264_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_2265_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_2266_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_2267_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2268_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_2269_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_2270_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2271_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_2272_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_2273_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2274_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_2275_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_2276_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_2277_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2278_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_2279_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2280_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_2281_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_2282_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_2283_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2284_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_2285_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_2286_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_2287_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_2288_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_2289_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_2290_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_less_numeral
thf(fact_2291_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_less_numeral
thf(fact_2292_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_less_numeral
thf(fact_2293_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_less_numeral
thf(fact_2294_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_2295_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_2296_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_2297_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_2298_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_2299_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2300_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_2301_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_2302_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_2303_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one_class.zero_le_one
thf(fact_2304_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_2305_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_2306_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_2307_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2308_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_2309_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_2310_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_2311_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2312_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_2313_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_2314_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_2315_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2316_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_2317_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_2318_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_2319_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2320_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_2321_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_2322_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_2323_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2324_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_2325_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_2326_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_2327_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_2328_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_2329_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_2330_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_2331_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2332_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_2333_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_2334_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_2335_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2336_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_2337_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_2338_vebt__succ_Osimps_I1_J,axiom,
    ! [B: $o,Uu2: $o] :
      ( ( B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_2339_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_2340_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2341_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_2342_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_2343_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_2344_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2345_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_2346_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_2347_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2348_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_2349_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_2350_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_2351_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2352_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_2353_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_2354_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2355_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_2356_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_2357_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2358_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_2359_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_2360_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_2361_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2362_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_2363_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_2364_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2365_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_2366_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_2367_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2368_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_2369_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_2370_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2371_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_2372_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_2373_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_2374_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2375_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_2376_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_2377_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_2378_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2379_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_2380_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_2381_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_2382_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_2383_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_2384_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_2385_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2386_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_2387_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_2388_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_2389_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_2390_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_2391_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_2392_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_2393_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_2394_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_2395_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_2396_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_2397_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2398_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_2399_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_2400_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_2401_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2402_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_2403_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2404_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_2405_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A5: real,B5: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B5 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_2406_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B5: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A5 @ B5 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_2407_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B5: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_2408_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_2409_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_2410_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2411_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2412_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_2413_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_2414_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2415_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2416_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_2417_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_2418_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_2419_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_2420_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_2421_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_2422_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_2423_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_2424_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2425_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_2426_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_2427_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C2 ) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_2428_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_2429_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_2430_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_2431_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_2432_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_2433_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_2434_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_2435_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_2436_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( A
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_2437_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_2438_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_2439_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_2440_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2441_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_2442_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2443_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_2444_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2445_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_2446_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2447_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_2448_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2449_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_2450_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2451_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_2452_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2453_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A5: real,B5: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B5 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_2454_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B5: rat] : ( ord_less_rat @ ( minus_minus_rat @ A5 @ B5 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_2455_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B5: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_2456_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_2457_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2458_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_2459_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2460_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_2461_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2462_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_2463_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_2464_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_2465_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_2466_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_2467_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_2468_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_2469_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_2470_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_2471_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_2472_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_2473_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_2474_zero__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2475_zero__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2476_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2477_zero__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2478_power__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono
thf(fact_2479_power__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono
thf(fact_2480_power__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).

% power_mono
thf(fact_2481_power__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono
thf(fact_2482_vebt__maxt_Osimps_I1_J,axiom,
    ! [B: $o,A: $o] :
      ( ( B
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_2483_zero__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2484_zero__less__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2485_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2486_zero__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2487_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X @ Y )
            = ( divide1717551699836669952omplex @ W @ Z ) )
          = ( ( times_times_complex @ X @ Z )
            = ( times_times_complex @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2488_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ( divide_divide_real @ X @ Y )
            = ( divide_divide_real @ W @ Z ) )
          = ( ( times_times_real @ X @ Z )
            = ( times_times_real @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2489_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X @ Y )
            = ( divide_divide_rat @ W @ Z ) )
          = ( ( times_times_rat @ X @ Z )
            = ( times_times_rat @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2490_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_2491_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_2492_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ C )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_2493_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_2494_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_2495_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_2496_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_2497_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_2498_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B
          = ( times_times_rat @ A @ C ) )
       => ( ( divide_divide_rat @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2499_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_2500_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_2501_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = B )
       => ( A
          = ( divide_divide_rat @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2502_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_2503_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_2504_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B @ C )
          = A )
        = ( B
          = ( times_times_rat @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2505_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_2506_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_2507_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A
          = ( divide_divide_rat @ B @ C ) )
        = ( ( times_times_rat @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2508_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_2509_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_2510_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A @ B )
          = one_one_rat )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2511_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_2512_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_2513_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_2514_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_2515_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_2516_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2517_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2518_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2519_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_2520_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_2521_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_2522_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ ( suc @ I3 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_2523_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M3: nat] :
            ( N
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_2524_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_2525_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M: nat] :
          ( N
          = ( suc @ M ) ) ) ).

% gr0_implies_Suc
thf(fact_2526_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( M2 = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M2
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_2527_one__is__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M2 @ N ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_2528_add__is__1,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_2529_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ~ ( P @ I4 ) )
            & ( P @ K ) ) ) ) ).

% ex_least_nat_le
thf(fact_2530_option_Osize_I4_J,axiom,
    ! [X2: product_prod_nat_nat] :
      ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2531_option_Osize_I4_J,axiom,
    ! [X2: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2532_option_Osize_I4_J,axiom,
    ! [X2: num] :
      ( ( size_size_option_num @ ( some_num @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2533_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K )
          & ( ( plus_plus_nat @ I @ K )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_2534_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2535_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2536_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2537_diff__less,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).

% diff_less
thf(fact_2538_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).

% mult_less_mono1
thf(fact_2539_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_2540_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ( times_times_nat @ K2 @ M2 )
          = ( times_times_nat @ K2 @ N ) )
        = ( M2 = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_2541_nat__mult__less__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
        = ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_2542_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_2543_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( divide_divide_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M2 @ N )
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_2544_diff__add__0,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_2545_nat__power__less__imp__less,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_2546_mult__eq__self__implies__10,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2
        = ( times_times_nat @ M2 @ N ) )
     => ( ( N = one_one_nat )
        | ( M2 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_2547_mod__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
          = N )
       => ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
         != N )
       => ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
          = ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ) ).

% mod_Suc
thf(fact_2548_mod__less__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).

% mod_less_divisor
thf(fact_2549_mod__eq__0D,axiom,
    ! [M2: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ D )
        = zero_zero_nat )
     => ? [Q2: nat] :
          ( M2
          = ( times_times_nat @ D @ Q2 ) ) ) ).

% mod_eq_0D
thf(fact_2550_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X5: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) @ X5 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X5: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X5 ) )
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ X5 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X5 ) ) ) ) ) ) ).

% vebt_insert.cases
thf(fact_2551_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o,Uw: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) )
     => ( ! [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,Va2: 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 @ Va2 @ Vb2 ) @ X5 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ X5 ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X5 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_2552_vebt__succ_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,B3: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat ) )
     => ( ! [Uv: $o,Uw: $o,N2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) )
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va2 ) )
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve ) )
           => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X5 ) ) ) ) ) ) ) ).

% vebt_succ.cases
thf(fact_2553_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X5: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ 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,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ X5 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_2554_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S2 ) @ X )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S2 ) ) ).

% vebt_insert.simps(2)
thf(fact_2555_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_2556_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv ) )
       => ( ! [Uu: $o] :
              ( X
             != ( vEBT_Leaf @ Uu @ $true ) )
         => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.cases
thf(fact_2557_set__update__memI,axiom,
    ! [N: nat,Xs2: list_real,X: real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_2558_set__update__memI,axiom,
    ! [N: nat,Xs2: list_complex,X: complex] :
      ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_2559_set__update__memI,axiom,
    ! [N: nat,Xs2: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
     => ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_2560_set__update__memI,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_2561_set__update__memI,axiom,
    ! [N: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_2562_set__update__memI,axiom,
    ! [N: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_2563_set__update__memI,axiom,
    ! [N: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_2564_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_VEBT_VEBT @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_2565_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ( list_update_o @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_o @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_2566_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( list_update_nat @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_nat @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_2567_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( list_update_int @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_int @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_2568_nth__list__update,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
            = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_2569_nth__list__update,axiom,
    ! [I: nat,Xs2: list_o,X: $o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ J )
        = ( ( ( I = J )
           => X )
          & ( ( I != J )
           => ( nth_o @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_2570_nth__list__update,axiom,
    ! [I: nat,Xs2: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
            = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_2571_nth__list__update,axiom,
    ! [I: nat,Xs2: list_int,J: nat,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ J )
            = ( nth_int @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_2572_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_2573_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ! [Uv: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv ) )
       => ( ! [Uu: $o] :
              ( X
             != ( vEBT_Leaf @ Uu @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_2574_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_2575_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2576_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_2577_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_2578_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_2579_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2580_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_2581_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_2582_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_2583_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2584_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_2585_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_2586_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_2587_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2588_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_2589_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_2590_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_2591_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2592_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_2593_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_2594_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2595_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_2596_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_2597_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2598_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_2599_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_2600_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2601_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_2602_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_2603_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_2604_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2605_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_2606_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_2607_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_2608_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2609_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_2610_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_2611_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2612_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_2613_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_2614_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_2615_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2616_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_2617_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_2618_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_2619_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2620_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_2621_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_2622_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2623_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_2624_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_2625_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_2626_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_2627_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_2628_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_2629_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_2630_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_2631_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_2632_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_2633_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_2634_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_2635_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_2636_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_2637_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_2638_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_2639_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_2640_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_2641_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2642_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_2643_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_2644_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_2645_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2646_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_2647_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_2648_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_2649_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E2 ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_2650_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_2651_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_2652_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_2653_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_2654_frac__le,axiom,
    ! [Y: real,X: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2655_frac__le,axiom,
    ! [Y: rat,X: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2656_frac__less,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2657_frac__less,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2658_frac__less2,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2659_frac__less2,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2660_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_2661_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_2662_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_2663_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_2664_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_2665_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2666_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_2667_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2668_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_2669_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_2670_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_2671_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2672_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_2673_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_2674_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_2675_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_eq_rat @ B @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_2676_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_2677_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_2678_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_2679_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2680_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_2681_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_2682_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2683_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_2684_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_2685_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2686_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_2687_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_2688_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2689_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_2690_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2691_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2692_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2693_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2694_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_2695_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_2696_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_2697_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_2698_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2699_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_2700_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_2701_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_2702_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_2703_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_2704_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2705_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2706_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_2707_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2708_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_2709_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2710_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_2711_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2712_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_2713_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2714_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_2715_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_2716_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_2717_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2718_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_2719_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2720_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_2721_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_2722_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_2723_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2724_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_2725_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2726_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_2727_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2728_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_2729_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_2730_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( X
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_2731_power__le__one,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_2732_power__le__one,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_2733_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_2734_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_2735_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2736_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( numeral_numeral_real @ W ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2737_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( numeral_numeral_rat @ W ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2738_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( ( numera6690914467698888265omplex @ W )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2739_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ( numeral_numeral_real @ W )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2740_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2741_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_2742_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_2743_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2744_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_2745_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_2746_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2747_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2748_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2749_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2750_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_2751_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_2752_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2753_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_2754_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_2755_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2756_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_2757_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_2758_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2759_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_2760_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_2761_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2762_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2763_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2764_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2765_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2766_power__inject__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ( power_power_real @ A @ ( suc @ N ) )
        = ( power_power_real @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2767_power__inject__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N ) )
        = ( power_power_rat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2768_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N ) )
        = ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2769_power__inject__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ( power_power_int @ A @ ( suc @ N ) )
        = ( power_power_int @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2770_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_2771_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_2772_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_2773_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_2774_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_2775_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_2776_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2777_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2778_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2779_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2780_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_2781_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_2782_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2783_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_2784_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_2785_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2786_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_2787_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_2788_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_2789_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_2790_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_2791_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_2792_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
      = zero_zero_nat ) ).

% cong_exp_iff_simps(1)
thf(fact_2793_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
      = zero_zero_int ) ).

% cong_exp_iff_simps(1)
thf(fact_2794_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_2795_num_Osize_I5_J,axiom,
    ! [X2: num] :
      ( ( size_size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_2796_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_nat @ K @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_2797_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_2798_one__less__mult,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% one_less_mult
thf(fact_2799_n__less__m__mult__n,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_2800_n__less__n__mult__m,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).

% n_less_n_mult_m
thf(fact_2801_length__pos__if__in__set,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2802_length__pos__if__in__set,axiom,
    ! [X: complex,Xs2: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2803_length__pos__if__in__set,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2804_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2805_length__pos__if__in__set,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2806_length__pos__if__in__set,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2807_length__pos__if__in__set,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2808_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_2809_nat__mult__le__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
        = ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_2810_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D2: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D2 ) )
                & ~ ( P @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_2811_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D2: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D2 ) )
           => ( P @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_2812_power__gt__expt,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ K2 @ ( power_power_nat @ N @ K2 ) ) ) ).

% power_gt_expt
thf(fact_2813_div__le__mono2,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K2 @ N ) @ ( divide_divide_nat @ K2 @ M2 ) ) ) ) ).

% div_le_mono2
thf(fact_2814_div__greater__zero__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ord_less_eq_nat @ N @ M2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% div_greater_zero_iff
thf(fact_2815_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).

% nat_one_le_power
thf(fact_2816_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q3 ) @ N )
        = ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q3 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_2817_nat__mult__div__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
        = ( divide_divide_nat @ M2 @ N ) ) ) ).

% nat_mult_div_cancel1
thf(fact_2818_div__eq__dividend__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ( divide_divide_nat @ M2 @ N )
          = M2 )
        = ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_2819_div__less__dividend,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).

% div_less_dividend
thf(fact_2820_mod__le__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).

% mod_le_divisor
thf(fact_2821_div__less__mono,axiom,
    ! [A4: nat,B4: nat,N: nat] :
      ( ( ord_less_nat @ A4 @ B4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A4 @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B4 @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A4 @ N ) @ ( divide_divide_nat @ B4 @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_2822_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) ) ).

% vebt_insert.simps(3)
thf(fact_2823_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_2824_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_2825_vebt__mint_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] :
          ( X
         != ( vEBT_Leaf @ A3 @ B3 ) )
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% vebt_mint.cases
thf(fact_2826_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y )
       => ( ( ? [Uv: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv ) )
           => Y )
         => ( ( ? [Uu: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu @ $true ) )
             => Y )
           => ( ( ? [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
               => ~ Y )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                 => Y ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_2827_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_2828_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2829_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_2830_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_2831_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2832_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_2833_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_2834_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2835_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_2836_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_2837_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2838_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_2839_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_2840_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2841_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_2842_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_2843_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2844_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_2845_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_2846_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2847_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_2848_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_2849_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2850_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_2851_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z3: real] :
          ( ( ord_less_real @ zero_zero_real @ Z3 )
         => ( ( ord_less_real @ Z3 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_2852_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z3 )
         => ( ( ord_less_rat @ Z3 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_2853_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_2854_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2855_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_2856_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2857_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_2858_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_2859_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_2860_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_2861_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_2862_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2863_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_2864_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2865_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_2866_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_2867_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_2868_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2869_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_2870_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2871_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_2872_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2873_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_2874_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_2875_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_2876_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2877_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_2878_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ( ord_less_eq_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2879_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_2880_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_2881_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_2882_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_2883_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_2884_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_2885_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_2886_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_2887_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_2888_power__Suc__less,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2889_power__Suc__less,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2890_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2891_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2892_power__Suc__le__self,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2893_power__Suc__le__self,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2894_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2895_power__Suc__le__self,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2896_power__Suc__less__one,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_2897_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_2898_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_2899_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_2900_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2901_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2902_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2903_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2904_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2905_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2906_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2907_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2908_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_2909_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_2910_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_2911_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_2912_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_2913_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( A3
                 => ( Y
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A3
                 => ( ( B3
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B3
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_2914_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( B3
                 => ( Y
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B3
                 => ( ( A3
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_2915_self__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2916_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2917_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2918_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2919_one__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2920_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2921_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2922_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2923_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B3 ) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A3 @ $true ) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
             => ( Y
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
               => ( Y
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                          & ~ ( ( Xa2 = Mi2 )
                              | ( Xa2 = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_2924_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_2925_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_2926_power__diff,axiom,
    ! [A: complex,N: nat,M2: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_2927_power__diff,axiom,
    ! [A: real,N: nat,M2: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_2928_power__diff,axiom,
    ! [A: rat,N: nat,M2: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_2929_power__diff,axiom,
    ! [A: nat,N: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_2930_power__diff,axiom,
    ! [A: int,N: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_2931_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_2932_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_2933_div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ~ ( ord_less_nat @ M2 @ N )
       => ( ( divide_divide_nat @ M2 @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).

% div_geq
thf(fact_2934_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M3: nat,N3: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M3 @ N3 )
            | ( N3 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).

% div_if
thf(fact_2935_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% add_eq_if
thf(fact_2936_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q3 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_2937_split__div,axiom,
    ! [P: nat > $o,M2: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) )
               => ( P @ I3 ) ) ) ) ) ) ).

% split_div
thf(fact_2938_dividend__less__div__times,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) ) ) ) ).

% dividend_less_div_times
thf(fact_2939_dividend__less__times__div,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_2940_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% mult_eq_if
thf(fact_2941_split__mod,axiom,
    ! [P: nat > $o,M2: nat,N: nat] :
      ( ( P @ ( modulo_modulo_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ M2 ) )
        & ( ( N != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I3 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_2942_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_2943_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_2944_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve2 )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_2945_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_2946_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X @ A )
     => ( ( ord_less_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2947_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_2948_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_2949_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_2950_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_2951_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_2952_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_2953_half__gt__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_2954_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_2955_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% half_gt_zero_iff
thf(fact_2956_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S2: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
       => ( ( ord_less_eq_real @ R2 @ S2 )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2957_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S2: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
       => ( ( ord_less_eq_rat @ R2 @ S2 )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2958_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_2959_zero__le__power2,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_2960_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_2961_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_2962_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2963_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_2964_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_2965_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_2966_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_2967_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_2968_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_2969_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_2970_power2__less__0,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).

% power2_less_0
thf(fact_2971_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_2972_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_2973_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_2974_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2975_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2976_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2977_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2978_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_2979_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_2980_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_2981_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_2982_less__2__cases,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N = zero_zero_nat )
        | ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_2983_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N = zero_zero_nat )
        | ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_2984_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P5: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2985_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P5: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2986_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P5: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2987_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P5: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2988_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P5: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2989_nat__induct2,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( P @ N2 )
             => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct2
thf(fact_2990_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_complex @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2991_power__minus__mult,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_real @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2992_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_rat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2993_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2994_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2995_le__div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( divide_divide_nat @ M2 @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).

% le_div_geq
thf(fact_2996_split__div_H,axiom,
    ! [P: nat > $o,M2: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q4: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M2 )
            & ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
            & ( P @ Q4 ) ) ) ) ).

% split_div'
thf(fact_2997_Suc__times__mod__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M2 @ N ) ) @ M2 )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_2998_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_2999_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_3000_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3001_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_3002_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_3003_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_3004_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3005_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_3006_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_3007_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3008_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_3009_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_3010_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_3011_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_3012_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_3013_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3014_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_3015_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_3016_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_3017_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_3018_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_3019_zero__le__even__power_H,axiom,
    ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_3020_zero__le__even__power_H,axiom,
    ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_3021_zero__le__even__power_H,axiom,
    ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_3022_nat__bit__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
       => ( ! [N2: nat] :
              ( ( P @ N2 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_bit_induct
thf(fact_3023_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_3024_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_3025_verit__le__mono__div,axiom,
    ! [A4: nat,B4: nat,N: nat] :
      ( ( ord_less_nat @ A4 @ B4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B4 @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B4 @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_3026_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
           => Y )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_3027_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A3 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B3 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_3028_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A3 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B3 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_3029_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_3030_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_3031_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3032_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3033_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3034_odd__power__less__zero,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_3035_odd__power__less__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_3036_odd__power__less__zero,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_3037_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: 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 ) ) @ TreeList @ 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 @ TreeList ) )
          & ~ ( ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 ) ) @ TreeList @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_3038_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_3039_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_3040_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A3 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B3 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_3041_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Uu: $o,Uv: $o] :
            ( X
           != ( vEBT_Leaf @ Uu @ Uv ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ 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 @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_3042_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => Y )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
               => ( Y
                  = ( ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ 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 @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_3043_mod__double__modulus,axiom,
    ! [M2: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo_modulo_nat @ X @ M2 ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3044_mod__double__modulus,axiom,
    ! [M2: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo_modulo_int @ X @ M2 ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3045_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_3046_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_3047_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => 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,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_3048_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A3 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B3 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
         => ( ! [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,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_3049_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_3050_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X @ Y ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_3051_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( ( vEBT_invar_vebt @ A1 @ A22 )
     => ( ( ? [A3: $o,B3: $o] :
              ( A1
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( A22
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( A22 = Deg2 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ( vEBT_invar_vebt @ X3 @ N2 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                     => ( ( M = N2 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N2 @ M ) )
                         => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
                ( ( A1
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( A22 = Deg2 )
                 => ( ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ( vEBT_invar_vebt @ X3 @ N2 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                       => ( ( M
                            = ( suc @ N2 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N2 @ M ) )
                           => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_invar_vebt @ X3 @ N2 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                         => ( ( M = N2 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N2 @ M ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X3: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                       => ~ ? [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 ) ) @ M ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                & ! [X3: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X3 )
                                                      & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X3: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                           => ( vEBT_invar_vebt @ X3 @ N2 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                           => ( ( M
                                = ( suc @ N2 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N2 @ M ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X3: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                         => ~ ? [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 ) ) @ M ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                  & ! [X3: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X3 )
                                                        & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_3052_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A12: vEBT_VEBT,A23: nat] :
          ( ( ? [A5: $o,B5: $o] :
                ( A12
                = ( vEBT_Leaf @ A5 @ B5 ) )
            & ( A23
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X4 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N3 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ N3 ) )
              & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X4 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
              & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X4 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N3 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ N3 ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
                      & ! [X4: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X4 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N3 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
                      & ! [X4: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_3053_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_3054_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_3055_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_3056_finite__Collect__le__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K2 ) ) ) ).

% finite_Collect_le_nat
thf(fact_3057_finite__Collect__less__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K2 ) ) ) ).

% finite_Collect_less_nat
thf(fact_3058_finite__Collect__bounded__ex,axiom,
    ! [P: real > $o,Q: real > real > $o] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X4: real] :
              ? [Y3: real] :
                ( ( P @ Y3 )
                & ( Q @ X4 @ Y3 ) ) ) )
        = ( ! [Y3: real] :
              ( ( P @ Y3 )
             => ( finite_finite_real
                @ ( collect_real
                  @ ^ [X4: real] : ( Q @ X4 @ Y3 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3059_finite__Collect__bounded__ex,axiom,
    ! [P: real > $o,Q: nat > real > $o] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X4: nat] :
              ? [Y3: real] :
                ( ( P @ Y3 )
                & ( Q @ X4 @ Y3 ) ) ) )
        = ( ! [Y3: real] :
              ( ( P @ Y3 )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X4: nat] : ( Q @ X4 @ Y3 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3060_finite__Collect__bounded__ex,axiom,
    ! [P: real > $o,Q: complex > real > $o] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [X4: complex] :
              ? [Y3: real] :
                ( ( P @ Y3 )
                & ( Q @ X4 @ Y3 ) ) ) )
        = ( ! [Y3: real] :
              ( ( P @ Y3 )
             => ( finite3207457112153483333omplex
                @ ( collect_complex
                  @ ^ [X4: complex] : ( Q @ X4 @ Y3 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3061_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: real > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X4: real] :
              ? [Y3: nat] :
                ( ( P @ Y3 )
                & ( Q @ X4 @ Y3 ) ) ) )
        = ( ! [Y3: nat] :
              ( ( P @ Y3 )
             => ( finite_finite_real
                @ ( collect_real
                  @ ^ [X4: real] : ( Q @ X4 @ Y3 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3062_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X4: nat] :
              ? [Y3: nat] :
                ( ( P @ Y3 )
                & ( Q @ X4 @ Y3 ) ) ) )
        = ( ! [Y3: nat] :
              ( ( P @ Y3 )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X4: nat] : ( Q @ X4 @ Y3 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3063_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: complex > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [X4: complex] :
              ? [Y3: nat] :
                ( ( P @ Y3 )
                & ( Q @ X4 @ Y3 ) ) ) )
        = ( ! [Y3: nat] :
              ( ( P @ Y3 )
             => ( finite3207457112153483333omplex
                @ ( collect_complex
                  @ ^ [X4: complex] : ( Q @ X4 @ Y3 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3064_finite__Collect__bounded__ex,axiom,
    ! [P: complex > $o,Q: real > complex > $o] :
      ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X4: real] :
              ? [Y3: complex] :
                ( ( P @ Y3 )
                & ( Q @ X4 @ Y3 ) ) ) )
        = ( ! [Y3: complex] :
              ( ( P @ Y3 )
             => ( finite_finite_real
                @ ( collect_real
                  @ ^ [X4: real] : ( Q @ X4 @ Y3 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3065_finite__Collect__bounded__ex,axiom,
    ! [P: complex > $o,Q: nat > complex > $o] :
      ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X4: nat] :
              ? [Y3: complex] :
                ( ( P @ Y3 )
                & ( Q @ X4 @ Y3 ) ) ) )
        = ( ! [Y3: complex] :
              ( ( P @ Y3 )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X4: nat] : ( Q @ X4 @ Y3 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3066_finite__Collect__bounded__ex,axiom,
    ! [P: complex > $o,Q: complex > complex > $o] :
      ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [X4: complex] :
              ? [Y3: complex] :
                ( ( P @ Y3 )
                & ( Q @ X4 @ Y3 ) ) ) )
        = ( ! [Y3: complex] :
              ( ( P @ Y3 )
             => ( finite3207457112153483333omplex
                @ ( collect_complex
                  @ ^ [X4: complex] : ( Q @ X4 @ Y3 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3067_finite__Collect__bounded__ex,axiom,
    ! [P: real > $o,Q: list_nat > real > $o] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite8100373058378681591st_nat
          @ ( collect_list_nat
            @ ^ [X4: list_nat] :
              ? [Y3: real] :
                ( ( P @ Y3 )
                & ( Q @ X4 @ Y3 ) ) ) )
        = ( ! [Y3: real] :
              ( ( P @ Y3 )
             => ( finite8100373058378681591st_nat
                @ ( collect_list_nat
                  @ ^ [X4: list_nat] : ( Q @ X4 @ Y3 ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3068_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z2: real] :
              ( ( power_power_real @ Z2 @ N )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_3069_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_3070_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ B3 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( B3
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B3
                     => ( Y = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv @ Uw ) )
               => ! [N2: nat] :
                    ( ( Xa2
                      = ( suc @ N2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Xa2 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ none_nat
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_3071_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3072_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3073_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3074_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3075_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3076_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3077_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3078_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3079_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3080_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3081_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3082_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3083_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_3084_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_3085_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ( ord_less_rat @ X @ Z )
        & ( ord_less_rat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3086_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z )
      = ( ( ord_less_num @ X @ Z )
        & ( ord_less_num @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3087_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_3088_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_3089_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_3090_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_3091_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_3092_i0__less,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
      = ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_3093_idiff__0__right,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
      = N ) ).

% idiff_0_right
thf(fact_3094_idiff__0,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_3095_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_3096_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_3097_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
      = ( ( ord_less_eq_num @ B @ A )
        & ( ord_less_eq_num @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_3098_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_3099_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_3100_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3101_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3102_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3103_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3104_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3105_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3106_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3107_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3108_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3109_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3110_finite__Diff2,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
        = ( finite3207457112153483333omplex @ A4 ) ) ) ).

% finite_Diff2
thf(fact_3111_finite__Diff2,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) )
        = ( finite_finite_nat @ A4 ) ) ) ).

% finite_Diff2
thf(fact_3112_finite__Diff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ).

% finite_Diff
thf(fact_3113_finite__Diff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% finite_Diff
thf(fact_3114_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
      = ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_3115_div__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( ord_less_int @ K2 @ L )
       => ( ( divide_divide_int @ K2 @ L )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_3116_div__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ K2 @ zero_zero_int )
     => ( ( ord_less_int @ L @ K2 )
       => ( ( divide_divide_int @ K2 @ L )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_3117_mod__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ K2 @ zero_zero_int )
     => ( ( ord_less_int @ L @ K2 )
       => ( ( modulo_modulo_int @ K2 @ L )
          = K2 ) ) ) ).

% mod_neg_neg_trivial
thf(fact_3118_mod__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( ord_less_int @ K2 @ L )
       => ( ( modulo_modulo_int @ K2 @ L )
          = K2 ) ) ) ).

% mod_pos_pos_trivial
thf(fact_3119_half__negative__int__iff,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% half_negative_int_iff
thf(fact_3120_half__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% half_nonnegative_int_iff
thf(fact_3121_set__bit__greater__eq,axiom,
    ! [K2: int,N: nat] : ( ord_less_eq_int @ K2 @ ( bit_se7879613467334960850it_int @ N @ K2 ) ) ).

% set_bit_greater_eq
thf(fact_3122_unset__bit__less__eq,axiom,
    ! [N: nat,K2: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K2 ) @ K2 ) ).

% unset_bit_less_eq
thf(fact_3123_int__div__less__self,axiom,
    ! [X: int,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ one_one_int @ K2 )
       => ( ord_less_int @ ( divide_divide_int @ X @ K2 ) @ X ) ) ) ).

% int_div_less_self
thf(fact_3124_neg__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_int @ L @ ( modulo_modulo_int @ K2 @ L ) ) ) ).

% neg_mod_bound
thf(fact_3125_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_int @ ( modulo_modulo_int @ K2 @ L ) @ L ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_3126_neg__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K2 @ L ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_3127_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K2 @ L ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_3128_mod__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K2 )
       => ( ( modulo_modulo_int @ K2 @ L )
          = ( modulo_modulo_int @ ( minus_minus_int @ K2 @ L ) @ L ) ) ) ) ).

% mod_pos_geq
thf(fact_3129_enat__0__less__mult__iff,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M2 @ N ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M2 )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).

% enat_0_less_mult_iff
thf(fact_3130_not__iless0,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_3131_iadd__is__0,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ M2 @ N )
        = zero_z5237406670263579293d_enat )
      = ( ( M2 = zero_z5237406670263579293d_enat )
        & ( N = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
thf(fact_3132_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_3133_i0__lb,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).

% i0_lb
thf(fact_3134_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
           => ( P @ M3 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X4 ) ) ) ) ).

% all_nat_less
thf(fact_3135_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
            & ( P @ M3 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X4 ) ) ) ) ).

% ex_nat_less
thf(fact_3136_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M7 )
     => ? [N2: nat] :
        ! [X3: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X3 @ M7 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X3 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_3137_finite__maxlen,axiom,
    ! [M7: set_list_o] :
      ( ( finite_finite_list_o @ M7 )
     => ? [N2: nat] :
        ! [X3: list_o] :
          ( ( member_list_o @ X3 @ M7 )
         => ( ord_less_nat @ ( size_size_list_o @ X3 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_3138_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M7 )
     => ? [N2: nat] :
        ! [X3: list_nat] :
          ( ( member_list_nat @ X3 @ M7 )
         => ( ord_less_nat @ ( size_size_list_nat @ X3 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_3139_finite__maxlen,axiom,
    ! [M7: set_list_int] :
      ( ( finite3922522038869484883st_int @ M7 )
     => ? [N2: nat] :
        ! [X3: list_int] :
          ( ( member_list_int @ X3 @ M7 )
         => ( ord_less_nat @ ( size_size_list_int @ X3 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_3140_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R3: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( ord_less_int @ B @ R3 )
           => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3141_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R3: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( ord_less_int @ R2 @ B )
           => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3142_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B2: int,Q5: int,R3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R3 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R3 ) @ zero_zero_int )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
           => ( ( ord_less_int @ zero_zero_int @ B2 )
             => ( ( ord_less_eq_int @ B2 @ B )
               => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3143_zdiv__mono2__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B2: int,Q5: int,R3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R3 ) )
       => ( ( ord_less_int @ R3 @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
           => ( ( ord_less_int @ zero_zero_int @ B2 )
             => ( ( ord_less_eq_int @ B2 @ B )
               => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3144_q__pos__lemma,axiom,
    ! [B2: int,Q5: int,R3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R3 ) )
     => ( ( ord_less_int @ R3 @ B2 )
       => ( ( ord_less_int @ zero_zero_int @ B2 )
         => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).

% q_pos_lemma
thf(fact_3145_zdiv__zmult2__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 ) ) ) ).

% zdiv_zmult2_eq
thf(fact_3146_div__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K2 )
       => ( ( divide_divide_int @ K2 @ L )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K2 @ L ) @ L ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_3147_zmod__eq__0__iff,axiom,
    ! [M2: int,D: int] :
      ( ( ( modulo_modulo_int @ M2 @ D )
        = zero_zero_int )
      = ( ? [Q4: int] :
            ( M2
            = ( times_times_int @ D @ Q4 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_3148_zmod__eq__0D,axiom,
    ! [M2: int,D: int] :
      ( ( ( modulo_modulo_int @ M2 @ D )
        = zero_zero_int )
     => ? [Q2: int] :
          ( M2
          = ( times_times_int @ D @ Q2 ) ) ) ).

% zmod_eq_0D
thf(fact_3149_mod__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K2 @ L ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K2 @ L )
          = ( plus_plus_int @ K2 @ L ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_3150_not__exp__less__eq__0__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_3151_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R4: real] :
          ( ( ord_less_real @ zero_zero_real @ R4 )
          & ( ( power_power_real @ R4 @ ( suc @ N ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_3152_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ one_one_real )
       => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).

% real_arch_pow_inv
thf(fact_3153_split__zdiv,axiom,
    ! [P: int > $o,N: int,K2: int] :
      ( ( P @ ( divide_divide_int @ N @ K2 ) )
      = ( ( ( K2 = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K2 )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K2 )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I3 ) @ J3 ) ) )
             => ( P @ I3 ) ) )
        & ( ( ord_less_int @ K2 @ zero_zero_int )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K2 @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I3 ) @ J3 ) ) )
             => ( P @ I3 ) ) ) ) ) ).

% split_zdiv
thf(fact_3154_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( divide_divide_int @ A @ B )
            = Q3 ) ) ) ) ).

% int_div_neg_eq
thf(fact_3155_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q3 ) ) ) ) ).

% int_div_pos_eq
thf(fact_3156_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_3157_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_3158_split__zmod,axiom,
    ! [P: int > $o,N: int,K2: int] :
      ( ( P @ ( modulo_modulo_int @ N @ K2 ) )
      = ( ( ( K2 = zero_zero_int )
         => ( P @ N ) )
        & ( ( ord_less_int @ zero_zero_int @ K2 )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K2 )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I3 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K2 @ zero_zero_int )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K2 @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I3 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_3159_verit__le__mono__div__int,axiom,
    ! [A4: int,B4: int,N: int] :
      ( ( ord_less_int @ A4 @ B4 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A4 @ N )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B4 @ N )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B4 @ N ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_3160_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ( ( power_power_real @ R4 @ N )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_3161_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X5: real] :
            ( ( ord_less_real @ zero_zero_real @ X5 )
            & ( ( power_power_real @ X5 @ N )
              = A )
            & ! [Y5: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y5 )
                  & ( ( power_power_real @ Y5 @ N )
                    = A ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_3162_split__neg__lemma,axiom,
    ! [K2: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ K2 @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N @ K2 ) @ ( modulo_modulo_int @ N @ K2 ) )
        = ( ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K2 @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I3 ) @ J3 ) ) )
             => ( P @ I3 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_3163_split__pos__lemma,axiom,
    ! [K2: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ( ( P @ ( divide_divide_int @ N @ K2 ) @ ( modulo_modulo_int @ N @ K2 ) )
        = ( ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K2 )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I3 ) @ J3 ) ) )
             => ( P @ I3 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_3164_zmod__zmult2__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 ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_3165_int__power__div__base,axiom,
    ! [M2: nat,K2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_int @ zero_zero_int @ K2 )
       => ( ( divide_divide_int @ ( power_power_int @ K2 @ M2 ) @ K2 )
          = ( power_power_int @ K2 @ ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).

% int_power_div_base
thf(fact_3166_finite__has__minimal2,axiom,
    ! [A4: set_real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ A @ A4 )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A4 )
            & ( ord_less_eq_real @ X5 @ A )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3167_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A @ A4 )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A4 )
            & ( ord_less_eq_set_nat @ X5 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3168_finite__has__minimal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A @ A4 )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A4 )
            & ( ord_less_eq_rat @ X5 @ A )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3169_finite__has__minimal2,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A @ A4 )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A4 )
            & ( ord_less_eq_num @ X5 @ A )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3170_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
            & ( ord_less_eq_nat @ X5 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3171_finite__has__minimal2,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A @ A4 )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A4 )
            & ( ord_less_eq_int @ X5 @ A )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3172_finite__has__maximal2,axiom,
    ! [A4: set_real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ A @ A4 )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A4 )
            & ( ord_less_eq_real @ A @ X5 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3173_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A @ A4 )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A4 )
            & ( ord_less_eq_set_nat @ A @ X5 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3174_finite__has__maximal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A @ A4 )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A4 )
            & ( ord_less_eq_rat @ A @ X5 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3175_finite__has__maximal2,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A @ A4 )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A4 )
            & ( ord_less_eq_num @ A @ X5 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3176_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
            & ( ord_less_eq_nat @ A @ X5 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3177_finite__has__maximal2,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A @ A4 )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A4 )
            & ( ord_less_eq_int @ A @ X5 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3178_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).

% neg_zdiv_mult_2
thf(fact_3179_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% pos_zdiv_mult_2
thf(fact_3180_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_3181_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_3182_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_3183_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D @ B )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_3184_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ( ord_less_eq_num @ D @ B )
       => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).

% max.mono
thf(fact_3185_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_3186_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_3187_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.orderE
thf(fact_3188_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_3189_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_3190_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_3191_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_3192_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) )
     => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).

% max.orderI
thf(fact_3193_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_3194_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_3195_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_3196_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_3197_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_3198_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_3199_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_num @ B @ A )
         => ~ ( ord_less_eq_num @ C @ A ) ) ) ).

% max.boundedE
thf(fact_3200_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_3201_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_3202_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_3203_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_3204_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ A )
       => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_3205_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_3206_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_3207_max_Oorder__iff,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B5: extended_enat,A5: extended_enat] :
          ( A5
          = ( ord_ma741700101516333627d_enat @ A5 @ B5 ) ) ) ) ).

% max.order_iff
thf(fact_3208_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B5: rat,A5: rat] :
          ( A5
          = ( ord_max_rat @ A5 @ B5 ) ) ) ) ).

% max.order_iff
thf(fact_3209_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B5: num,A5: num] :
          ( A5
          = ( ord_max_num @ A5 @ B5 ) ) ) ) ).

% max.order_iff
thf(fact_3210_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B5: nat,A5: nat] :
          ( A5
          = ( ord_max_nat @ A5 @ B5 ) ) ) ) ).

% max.order_iff
thf(fact_3211_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B5: int,A5: int] :
          ( A5
          = ( ord_max_int @ A5 @ B5 ) ) ) ) ).

% max.order_iff
thf(fact_3212_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded1
thf(fact_3213_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_3214_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_3215_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_3216_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_3217_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded2
thf(fact_3218_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_3219_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_3220_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_3221_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_3222_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_3223_le__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_eq_rat @ Z @ X )
        | ( ord_less_eq_rat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3224_le__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_eq_num @ Z @ X )
        | ( ord_less_eq_num @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3225_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_3226_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_3227_max_Oabsorb__iff1,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B5: extended_enat,A5: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A5 @ B5 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_3228_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B5: rat,A5: rat] :
          ( ( ord_max_rat @ A5 @ B5 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_3229_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B5: num,A5: num] :
          ( ( ord_max_num @ A5 @ B5 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_3230_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( ord_max_nat @ A5 @ B5 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_3231_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B5: int,A5: int] :
          ( ( ord_max_int @ A5 @ B5 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_3232_max_Oabsorb__iff2,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A5: extended_enat,B5: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A5 @ B5 )
          = B5 ) ) ) ).

% max.absorb_iff2
thf(fact_3233_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B5: rat] :
          ( ( ord_max_rat @ A5 @ B5 )
          = B5 ) ) ) ).

% max.absorb_iff2
thf(fact_3234_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A5: num,B5: num] :
          ( ( ord_max_num @ A5 @ B5 )
          = B5 ) ) ) ).

% max.absorb_iff2
thf(fact_3235_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( ord_max_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% max.absorb_iff2
thf(fact_3236_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B5: int] :
          ( ( ord_max_int @ A5 @ B5 )
          = B5 ) ) ) ).

% max.absorb_iff2
thf(fact_3237_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_3238_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_3239_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_3240_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_3241_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_3242_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_3243_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_3244_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_eq_num @ C @ B )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_3245_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_3246_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_3247_Diff__infinite__finite,axiom,
    ! [T3: set_complex,S3: set_complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S3 )
       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_3248_Diff__infinite__finite,axiom,
    ! [T3: set_nat,S3: set_nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ~ ( finite_finite_nat @ S3 )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_3249_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_3250_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_3251_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Z @ X )
        | ( ord_less_rat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3252_less__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_num @ Z @ X )
        | ( ord_less_num @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3253_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_3254_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_3255_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_3256_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_3257_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_3258_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_num @ B @ A )
         => ~ ( ord_less_num @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_3259_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_3260_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_3261_max_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B5: extended_enat,A5: extended_enat] :
          ( ( A5
            = ( ord_ma741700101516333627d_enat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3262_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B5: real,A5: real] :
          ( ( A5
            = ( ord_max_real @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3263_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B5: rat,A5: rat] :
          ( ( A5
            = ( ord_max_rat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3264_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B5: num,A5: num] :
          ( ( A5
            = ( ord_max_num @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3265_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A5: nat] :
          ( ( A5
            = ( ord_max_nat @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3266_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B5: int,A5: int] :
          ( ( A5
            = ( ord_max_int @ A5 @ B5 ) )
          & ( A5 != B5 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3267_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_3268_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_3269_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_3270_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_num @ C @ A )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_3271_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_3272_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_3273_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_3274_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_3275_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ B )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_3276_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_num @ C @ B )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_3277_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_3278_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_3279_finite__image__set,axiom,
    ! [P: real > $o,F: real > real] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Uu3: real] :
            ? [X4: real] :
              ( ( Uu3
                = ( F @ X4 ) )
              & ( P @ X4 ) ) ) ) ) ).

% finite_image_set
thf(fact_3280_finite__image__set,axiom,
    ! [P: real > $o,F: real > nat] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu3: nat] :
            ? [X4: real] :
              ( ( Uu3
                = ( F @ X4 ) )
              & ( P @ X4 ) ) ) ) ) ).

% finite_image_set
thf(fact_3281_finite__image__set,axiom,
    ! [P: real > $o,F: real > complex] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Uu3: complex] :
            ? [X4: real] :
              ( ( Uu3
                = ( F @ X4 ) )
              & ( P @ X4 ) ) ) ) ) ).

% finite_image_set
thf(fact_3282_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Uu3: real] :
            ? [X4: nat] :
              ( ( Uu3
                = ( F @ X4 ) )
              & ( P @ X4 ) ) ) ) ) ).

% finite_image_set
thf(fact_3283_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu3: nat] :
            ? [X4: nat] :
              ( ( Uu3
                = ( F @ X4 ) )
              & ( P @ X4 ) ) ) ) ) ).

% finite_image_set
thf(fact_3284_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Uu3: complex] :
            ? [X4: nat] :
              ( ( Uu3
                = ( F @ X4 ) )
              & ( P @ X4 ) ) ) ) ) ).

% finite_image_set
thf(fact_3285_finite__image__set,axiom,
    ! [P: complex > $o,F: complex > real] :
      ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Uu3: real] :
            ? [X4: complex] :
              ( ( Uu3
                = ( F @ X4 ) )
              & ( P @ X4 ) ) ) ) ) ).

% finite_image_set
thf(fact_3286_finite__image__set,axiom,
    ! [P: complex > $o,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu3: nat] :
            ? [X4: complex] :
              ( ( Uu3
                = ( F @ X4 ) )
              & ( P @ X4 ) ) ) ) ) ).

% finite_image_set
thf(fact_3287_finite__image__set,axiom,
    ! [P: complex > $o,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Uu3: complex] :
            ? [X4: complex] :
              ( ( Uu3
                = ( F @ X4 ) )
              & ( P @ X4 ) ) ) ) ) ).

% finite_image_set
thf(fact_3288_finite__image__set,axiom,
    ! [P: real > $o,F: real > list_nat] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Uu3: list_nat] :
            ? [X4: real] :
              ( ( Uu3
                = ( F @ X4 ) )
              & ( P @ X4 ) ) ) ) ) ).

% finite_image_set
thf(fact_3289_finite__image__set2,axiom,
    ! [P: real > $o,Q: real > $o,F: real > real > real] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real @ ( collect_real @ Q ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Uu3: real] :
              ? [X4: real,Y3: real] :
                ( ( Uu3
                  = ( F @ X4 @ Y3 ) )
                & ( P @ X4 )
                & ( Q @ Y3 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3290_finite__image__set2,axiom,
    ! [P: real > $o,Q: real > $o,F: real > real > nat] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real @ ( collect_real @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu3: nat] :
              ? [X4: real,Y3: real] :
                ( ( Uu3
                  = ( F @ X4 @ Y3 ) )
                & ( P @ X4 )
                & ( Q @ Y3 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3291_finite__image__set2,axiom,
    ! [P: real > $o,Q: real > $o,F: real > real > complex] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real @ ( collect_real @ Q ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Uu3: complex] :
              ? [X4: real,Y3: real] :
                ( ( Uu3
                  = ( F @ X4 @ Y3 ) )
                & ( P @ X4 )
                & ( Q @ Y3 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3292_finite__image__set2,axiom,
    ! [P: real > $o,Q: nat > $o,F: real > nat > real] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Uu3: real] :
              ? [X4: real,Y3: nat] :
                ( ( Uu3
                  = ( F @ X4 @ Y3 ) )
                & ( P @ X4 )
                & ( Q @ Y3 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3293_finite__image__set2,axiom,
    ! [P: real > $o,Q: nat > $o,F: real > nat > nat] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu3: nat] :
              ? [X4: real,Y3: nat] :
                ( ( Uu3
                  = ( F @ X4 @ Y3 ) )
                & ( P @ X4 )
                & ( Q @ Y3 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3294_finite__image__set2,axiom,
    ! [P: real > $o,Q: nat > $o,F: real > nat > complex] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Uu3: complex] :
              ? [X4: real,Y3: nat] :
                ( ( Uu3
                  = ( F @ X4 @ Y3 ) )
                & ( P @ X4 )
                & ( Q @ Y3 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3295_finite__image__set2,axiom,
    ! [P: real > $o,Q: complex > $o,F: real > complex > real] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite3207457112153483333omplex @ ( collect_complex @ Q ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Uu3: real] :
              ? [X4: real,Y3: complex] :
                ( ( Uu3
                  = ( F @ X4 @ Y3 ) )
                & ( P @ X4 )
                & ( Q @ Y3 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3296_finite__image__set2,axiom,
    ! [P: real > $o,Q: complex > $o,F: real > complex > nat] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite3207457112153483333omplex @ ( collect_complex @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu3: nat] :
              ? [X4: real,Y3: complex] :
                ( ( Uu3
                  = ( F @ X4 @ Y3 ) )
                & ( P @ X4 )
                & ( Q @ Y3 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3297_finite__image__set2,axiom,
    ! [P: real > $o,Q: complex > $o,F: real > complex > complex] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite3207457112153483333omplex @ ( collect_complex @ Q ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Uu3: complex] :
              ? [X4: real,Y3: complex] :
                ( ( Uu3
                  = ( F @ X4 @ Y3 ) )
                & ( P @ X4 )
                & ( Q @ Y3 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3298_finite__image__set2,axiom,
    ! [P: nat > $o,Q: real > $o,F: nat > real > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_real @ ( collect_real @ Q ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Uu3: real] :
              ? [X4: nat,Y3: real] :
                ( ( Uu3
                  = ( F @ X4 @ Y3 ) )
                & ( P @ X4 )
                & ( Q @ Y3 ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3299_finite__has__minimal,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A4 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3300_finite__has__minimal,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3301_finite__has__minimal,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A4 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3302_finite__has__minimal,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A4 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3303_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3304_finite__has__minimal,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A4 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3305_finite__has__maximal,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A4 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3306_finite__has__maximal,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3307_finite__has__maximal,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A4 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3308_finite__has__maximal,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A4 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3309_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3310_finite__has__maximal,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A4 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3311_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B3 ) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A3 @ $true ) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
               => ( ( Y
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ Xa2 ) ) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                              & ~ ( ( Xa2 = Mi2 )
                                  | ( Xa2 = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_3312_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A3 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B3 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa2 ) ) )
           => ( ! [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,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_3313_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa2 ) ) ) )
           => ( ! [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,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_3314_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A3 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B3 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Xa2 ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_3315_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa2 ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_3316_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Xa2 ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_3317_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_3318_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 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ 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,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ 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 @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_3319_max__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q3 @ zero_z5237406670263579293d_enat )
      = Q3 ) ).

% max_enat_simps(2)
thf(fact_3320_max__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q3 )
      = Q3 ) ).

% max_enat_simps(3)
thf(fact_3321_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K2 ) @ zero_zero_int )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_3322_set__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K2 ) @ zero_zero_int )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_3323_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K2 ) @ zero_zero_int )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_3324_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_3325_imult__is__0,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ M2 @ N )
        = zero_z5237406670263579293d_enat )
      = ( ( M2 = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
thf(fact_3326_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ 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 @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_3327_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 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( 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,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
                 => ( ( Y
                      = ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ 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 @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_3328_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W @ Z ) ) ).

% zle_diff1_eq
thf(fact_3329_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% zle_add1_eq_le
thf(fact_3330_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_3331_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_3332_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_3333_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_3334_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_3335_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_3336_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_3337_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_3338_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3339_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3340_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_3341_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_3342_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_3343_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_3344_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3345_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3346_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_3347_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_3348_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_3349_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_3350_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3351_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3352_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_3353_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_3354_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_3355_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_3356_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_3357_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( plus_plus_rat @ A @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
thf(fact_3358_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_3359_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I )
        & ( ord_less_eq_set_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3360_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I )
        & ( ord_less_eq_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3361_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I )
        & ( ord_less_eq_num @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3362_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I )
        & ( ord_less_eq_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3363_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I )
        & ( ord_less_eq_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3364_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
      = ( ( ord_less_eq_real @ L @ I )
        & ( ord_less_eq_real @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3365_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_3366_Icc__eq__Icc,axiom,
    ! [L: rat,H2: rat,L3: rat,H3: rat] :
      ( ( ( set_or633870826150836451st_rat @ L @ H2 )
        = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_rat @ L @ H2 )
          & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3367_Icc__eq__Icc,axiom,
    ! [L: num,H2: num,L3: num,H3: num] :
      ( ( ( set_or7049704709247886629st_num @ L @ H2 )
        = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_num @ L @ H2 )
          & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3368_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_3369_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_3370_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_3371_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M7: nat] :
      ( ( P @ X )
     => ( ! [X5: nat] :
            ( ( P @ X5 )
           => ( ord_less_eq_nat @ X5 @ M7 ) )
       => ~ ! [M: nat] :
              ( ( P @ M )
             => ~ ! [X3: nat] :
                    ( ( P @ X3 )
                   => ( ord_less_eq_nat @ X3 @ M ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_3372_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc3368934014287244435at_num] :
      ~ ! [F2: nat > num > num,A3: nat,B3: nat,Acc: num] :
          ( X
         != ( produc851828971589881931at_num @ F2 @ ( produc1195630363706982562at_num @ A3 @ ( product_Pair_nat_num @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_3373_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F2: nat > nat > nat,A3: nat,B3: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_3374_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_3375_times__int__code_I1_J,axiom,
    ! [K2: int] :
      ( ( times_times_int @ K2 @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_3376_minus__int__code_I1_J,axiom,
    ! [K2: int] :
      ( ( minus_minus_int @ K2 @ zero_zero_int )
      = K2 ) ).

% minus_int_code(1)
thf(fact_3377_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
      = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_3378_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(1)
thf(fact_3379_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
      = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_3380_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(3)
thf(fact_3381_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M3: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N6 )
         => ( ord_less_nat @ X4 @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_3382_bounded__nat__set__is__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ N5 )
         => ( ord_less_nat @ X5 @ N ) )
     => ( finite_finite_nat @ N5 ) ) ).

% bounded_nat_set_is_finite
thf(fact_3383_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M3: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N6 )
         => ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_3384_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_3385_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_3386_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_3387_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_3388_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K2: int] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_int @ zero_zero_int @ K2 )
       => ( ord_less_int @ ( times_times_int @ K2 @ I ) @ ( times_times_int @ K2 @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_3389_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_3390_int__ge__induct,axiom,
    ! [K2: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K2 @ I )
     => ( ( P @ K2 )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ K2 @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_3391_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ( ord_less_int @ W @ Z )
        | ( W = Z ) ) ) ).

% zless_add1_eq
thf(fact_3392_int__gr__induct,axiom,
    ! [K2: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K2 @ I )
     => ( ( P @ ( plus_plus_int @ K2 @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ K2 @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_3393_int__le__induct,axiom,
    ! [I: int,K2: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K2 )
     => ( ( P @ K2 )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K2 )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_3394_int__less__induct,axiom,
    ! [I: int,K2: int,P: int > $o] :
      ( ( ord_less_int @ I @ K2 )
     => ( ( P @ ( minus_minus_int @ K2 @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ I2 @ K2 )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_3395_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_3396_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B )
          | ( ( ord_less_eq_rat @ C @ A )
            & ( ord_less_eq_rat @ B @ D )
            & ( ( ord_less_rat @ C @ A )
              | ( ord_less_rat @ B @ D ) ) ) )
        & ( ord_less_eq_rat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3397_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B )
          | ( ( ord_less_eq_num @ C @ A )
            & ( ord_less_eq_num @ B @ D )
            & ( ( ord_less_num @ C @ A )
              | ( ord_less_num @ B @ D ) ) ) )
        & ( ord_less_eq_num @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3398_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_3399_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_3400_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_3401_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_3402_pos__zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M2 )
     => ( ( ( times_times_int @ M2 @ N )
          = one_one_int )
        = ( ( M2 = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_3403_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_3404_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ Z )
     => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).

% zless_imp_add1_zle
thf(fact_3405_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
      = ( ord_less_int @ W @ Z ) ) ).

% add1_zle_eq
thf(fact_3406_int__induct,axiom,
    ! [P: int > $o,K2: int,I: int] :
      ( ( P @ K2 )
     => ( ! [I2: int] :
            ( ( ord_less_eq_int @ K2 @ I2 )
           => ( ( P @ I2 )
             => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K2 )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_3407_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_3408_cppi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z5 @ X5 )
           => ( ( P @ X5 )
              = ( P6 @ X5 ) ) )
       => ( ! [X5: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb: int] :
                      ( ( member_int @ Xb @ A4 )
                     => ( X5
                       != ( minus_minus_int @ Xb @ Xa ) ) ) )
             => ( ( P @ X5 )
               => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
         => ( ! [X5: int,K: int] :
                ( ( P6 @ X5 )
                = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D4 ) ) ) )
           => ( ( ? [X6: int] : ( P @ X6 ) )
              = ( ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X4 ) )
                | ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y3: int] :
                        ( ( member_int @ Y3 @ A4 )
                        & ( P @ ( minus_minus_int @ Y3 @ X4 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_3409_cpmi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z5 )
           => ( ( P @ X5 )
              = ( P6 @ X5 ) ) )
       => ( ! [X5: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb: int] :
                      ( ( member_int @ Xb @ B4 )
                     => ( X5
                       != ( plus_plus_int @ Xb @ Xa ) ) ) )
             => ( ( P @ X5 )
               => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
         => ( ! [X5: int,K: int] :
                ( ( P6 @ X5 )
                = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D4 ) ) ) )
           => ( ( ? [X6: int] : ( P @ X6 ) )
              = ( ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X4 ) )
                | ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y3: int] :
                        ( ( member_int @ Y3 @ B4 )
                        & ( P @ ( plus_plus_int @ Y3 @ X4 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_3410_aset_I8_J,axiom,
    ! [D4: int,A4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A4 )
                 => ( X3
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X3 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_3411_aset_I6_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X3
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X3 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_3412_bset_I8_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B4 )
                   => ( X3
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X3 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_3413_bset_I6_J,axiom,
    ! [D4: int,B4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B4 )
                 => ( X3
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X3 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_3414_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_3415_aset_I7_J,axiom,
    ! [D4: int,A4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A4 )
                 => ( X3
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X3 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_3416_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z3 )
     => ~ ( ord_less_real @ T @ X3 ) ) ).

% minf(7)
thf(fact_3417_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z3 )
     => ~ ( ord_less_rat @ T @ X3 ) ) ).

% minf(7)
thf(fact_3418_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( ( ord_less_num @ X3 @ Z3 )
     => ~ ( ord_less_num @ T @ X3 ) ) ).

% minf(7)
thf(fact_3419_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ~ ( ord_less_nat @ T @ X3 ) ) ).

% minf(7)
thf(fact_3420_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z3 )
     => ~ ( ord_less_int @ T @ X3 ) ) ).

% minf(7)
thf(fact_3421_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z3 )
     => ( ord_less_real @ X3 @ T ) ) ).

% minf(5)
thf(fact_3422_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z3 )
     => ( ord_less_rat @ X3 @ T ) ) ).

% minf(5)
thf(fact_3423_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( ( ord_less_num @ X3 @ Z3 )
     => ( ord_less_num @ X3 @ T ) ) ).

% minf(5)
thf(fact_3424_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ( ord_less_nat @ X3 @ T ) ) ).

% minf(5)
thf(fact_3425_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z3 )
     => ( ord_less_int @ X3 @ T ) ) ).

% minf(5)
thf(fact_3426_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z3 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_3427_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z3 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_3428_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( ( ord_less_num @ X3 @ Z3 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_3429_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_3430_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z3 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_3431_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z3 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_3432_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z3 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_3433_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( ( ord_less_num @ X3 @ Z3 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_3434_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_3435_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z3 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_3436_minf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z5: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3437_minf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z5: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3438_minf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z5: num] :
        ! [X5: num] :
          ( ( ord_less_num @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3439_minf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z5: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3440_minf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z5: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3441_minf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z5: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3442_minf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z5: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3443_minf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z5: num] :
        ! [X5: num] :
          ( ( ord_less_num @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3444_minf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z5: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3445_minf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z5: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3446_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z3 @ X3 )
     => ( ord_less_real @ T @ X3 ) ) ).

% pinf(7)
thf(fact_3447_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z3 @ X3 )
     => ( ord_less_rat @ T @ X3 ) ) ).

% pinf(7)
thf(fact_3448_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( ( ord_less_num @ Z3 @ X3 )
     => ( ord_less_num @ T @ X3 ) ) ).

% pinf(7)
thf(fact_3449_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ( ord_less_nat @ T @ X3 ) ) ).

% pinf(7)
thf(fact_3450_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z3 @ X3 )
     => ( ord_less_int @ T @ X3 ) ) ).

% pinf(7)
thf(fact_3451_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z3 @ X3 )
     => ~ ( ord_less_real @ X3 @ T ) ) ).

% pinf(5)
thf(fact_3452_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z3 @ X3 )
     => ~ ( ord_less_rat @ X3 @ T ) ) ).

% pinf(5)
thf(fact_3453_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( ( ord_less_num @ Z3 @ X3 )
     => ~ ( ord_less_num @ X3 @ T ) ) ).

% pinf(5)
thf(fact_3454_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ~ ( ord_less_nat @ X3 @ T ) ) ).

% pinf(5)
thf(fact_3455_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z3 @ X3 )
     => ~ ( ord_less_int @ X3 @ T ) ) ).

% pinf(5)
thf(fact_3456_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z3 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_3457_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z3 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_3458_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( ( ord_less_num @ Z3 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_3459_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_3460_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z3 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_3461_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z3 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_3462_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z3 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_3463_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( ( ord_less_num @ Z3 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_3464_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_3465_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z3 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_3466_pinf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z5: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z3 @ X3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3467_pinf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z5: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z3 @ X3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3468_pinf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z5: num] :
        ! [X5: num] :
          ( ( ord_less_num @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z3 @ X3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3469_pinf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z5: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z3 @ X3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3470_pinf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z5: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z3 @ X3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3471_pinf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z5: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z3 @ X3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3472_pinf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z5: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z3 @ X3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3473_pinf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z5: num] :
        ! [X5: num] :
          ( ( ord_less_num @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z3 @ X3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3474_pinf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z5: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z3 @ X3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3475_pinf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z5: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z3: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z3 @ X3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3476_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z3 )
     => ~ ( ord_less_eq_real @ T @ X3 ) ) ).

% minf(8)
thf(fact_3477_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z3 )
     => ~ ( ord_less_eq_rat @ T @ X3 ) ) ).

% minf(8)
thf(fact_3478_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( ( ord_less_num @ X3 @ Z3 )
     => ~ ( ord_less_eq_num @ T @ X3 ) ) ).

% minf(8)
thf(fact_3479_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X3 ) ) ).

% minf(8)
thf(fact_3480_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z3 )
     => ~ ( ord_less_eq_int @ T @ X3 ) ) ).

% minf(8)
thf(fact_3481_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z3 )
     => ( ord_less_eq_real @ X3 @ T ) ) ).

% minf(6)
thf(fact_3482_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z3 )
     => ( ord_less_eq_rat @ X3 @ T ) ) ).

% minf(6)
thf(fact_3483_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( ( ord_less_num @ X3 @ Z3 )
     => ( ord_less_eq_num @ X3 @ T ) ) ).

% minf(6)
thf(fact_3484_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ( ord_less_eq_nat @ X3 @ T ) ) ).

% minf(6)
thf(fact_3485_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z3 )
     => ( ord_less_eq_int @ X3 @ T ) ) ).

% minf(6)
thf(fact_3486_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z3 @ X3 )
     => ( ord_less_eq_real @ T @ X3 ) ) ).

% pinf(8)
thf(fact_3487_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z3 @ X3 )
     => ( ord_less_eq_rat @ T @ X3 ) ) ).

% pinf(8)
thf(fact_3488_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( ( ord_less_num @ Z3 @ X3 )
     => ( ord_less_eq_num @ T @ X3 ) ) ).

% pinf(8)
thf(fact_3489_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ( ord_less_eq_nat @ T @ X3 ) ) ).

% pinf(8)
thf(fact_3490_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z3 @ X3 )
     => ( ord_less_eq_int @ T @ X3 ) ) ).

% pinf(8)
thf(fact_3491_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z3 @ X3 )
     => ~ ( ord_less_eq_real @ X3 @ T ) ) ).

% pinf(6)
thf(fact_3492_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z3 @ X3 )
     => ~ ( ord_less_eq_rat @ X3 @ T ) ) ).

% pinf(6)
thf(fact_3493_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( ( ord_less_num @ Z3 @ X3 )
     => ~ ( ord_less_eq_num @ X3 @ T ) ) ).

% pinf(6)
thf(fact_3494_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ~ ( ord_less_eq_nat @ X3 @ T ) ) ).

% pinf(6)
thf(fact_3495_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z3 @ X3 )
     => ~ ( ord_less_eq_int @ X3 @ T ) ) ).

% pinf(6)
thf(fact_3496_inf__period_I2_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X5: real,K: real] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K @ D4 ) ) ) )
     => ( ! [X5: real,K: real] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K @ D4 ) ) ) )
       => ! [X3: real,K4: real] :
            ( ( ( P @ X3 )
              | ( Q @ X3 ) )
            = ( ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3497_inf__period_I2_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X5: rat,K: rat] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K @ D4 ) ) ) )
     => ( ! [X5: rat,K: rat] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K @ D4 ) ) ) )
       => ! [X3: rat,K4: rat] :
            ( ( ( P @ X3 )
              | ( Q @ X3 ) )
            = ( ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3498_inf__period_I2_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X5: int,K: int] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D4 ) ) ) )
     => ( ! [X5: int,K: int] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D4 ) ) ) )
       => ! [X3: int,K4: int] :
            ( ( ( P @ X3 )
              | ( Q @ X3 ) )
            = ( ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3499_inf__period_I1_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X5: real,K: real] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K @ D4 ) ) ) )
     => ( ! [X5: real,K: real] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K @ D4 ) ) ) )
       => ! [X3: real,K4: real] :
            ( ( ( P @ X3 )
              & ( Q @ X3 ) )
            = ( ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3500_inf__period_I1_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X5: rat,K: rat] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K @ D4 ) ) ) )
     => ( ! [X5: rat,K: rat] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K @ D4 ) ) ) )
       => ! [X3: rat,K4: rat] :
            ( ( ( P @ X3 )
              & ( Q @ X3 ) )
            = ( ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3501_inf__period_I1_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X5: int,K: int] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D4 ) ) ) )
     => ( ! [X5: int,K: int] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D4 ) ) ) )
       => ! [X3: int,K4: int] :
            ( ( ( P @ X3 )
              & ( Q @ X3 ) )
            = ( ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3502_bset_I1_J,axiom,
    ! [D4: int,B4: set_int,P: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ B4 )
                 => ( X5
                   != ( plus_plus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X5 )
           => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
     => ( ! [X5: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ B4 )
                   => ( X5
                     != ( plus_plus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X5 )
             => ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B4 )
                   => ( X3
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
             => ( ( P @ ( minus_minus_int @ X3 @ D4 ) )
                & ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_3503_bset_I2_J,axiom,
    ! [D4: int,B4: set_int,P: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ B4 )
                 => ( X5
                   != ( plus_plus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X5 )
           => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
     => ( ! [X5: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ B4 )
                   => ( X5
                     != ( plus_plus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X5 )
             => ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B4 )
                   => ( X3
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
             => ( ( P @ ( minus_minus_int @ X3 @ D4 ) )
                | ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_3504_aset_I1_J,axiom,
    ! [D4: int,A4: set_int,P: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ A4 )
                 => ( X5
                   != ( minus_minus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X5 )
           => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
     => ( ! [X5: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ A4 )
                   => ( X5
                     != ( minus_minus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X5 )
             => ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X3
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
             => ( ( P @ ( plus_plus_int @ X3 @ D4 ) )
                & ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_3505_aset_I2_J,axiom,
    ! [D4: int,A4: set_int,P: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ A4 )
                 => ( X5
                   != ( minus_minus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X5 )
           => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
     => ( ! [X5: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ A4 )
                   => ( X5
                     != ( minus_minus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X5 )
             => ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X3
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
             => ( ( P @ ( plus_plus_int @ X3 @ D4 ) )
                | ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_3506_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int,K: int] :
            ( ( P1 @ X5 )
            = ( P1 @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X5: int] :
              ( ( ord_less_int @ X5 @ Z5 )
             => ( ( P @ X5 )
                = ( P1 @ X5 ) ) )
         => ( ? [X_1: int] : ( P1 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% minusinfinity
thf(fact_3507_plusinfinity,axiom,
    ! [D: int,P6: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int,K: int] :
            ( ( P6 @ X5 )
            = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X5: int] :
              ( ( ord_less_int @ Z5 @ X5 )
             => ( ( P @ X5 )
                = ( P6 @ X5 ) ) )
         => ( ? [X_1: int] : ( P6 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% plusinfinity
thf(fact_3508_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int] :
            ( ( P @ X5 )
           => ( P @ ( plus_plus_int @ X5 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K2 )
         => ! [X3: int] :
              ( ( P @ X3 )
             => ( P @ ( plus_plus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3509_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int] :
            ( ( P @ X5 )
           => ( P @ ( minus_minus_int @ X5 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K2 )
         => ! [X3: int] :
              ( ( P @ X3 )
             => ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_3510_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int,K: int] :
            ( ( P @ X5 )
            = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) )
       => ( ( ? [X6: int] : ( P @ X6 ) )
          = ( ? [X4: int] :
                ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X4 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_3511_bset_I3_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B4 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B4 )
                   => ( X3
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X3 = T )
             => ( ( minus_minus_int @ X3 @ D4 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_3512_bset_I4_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B4 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B4 )
                   => ( X3
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X3 != T )
             => ( ( minus_minus_int @ X3 @ D4 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_3513_bset_I5_J,axiom,
    ! [D4: int,B4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B4 )
                 => ( X3
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X3 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_3514_bset_I7_J,axiom,
    ! [D4: int,T: int,B4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B4 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B4 )
                   => ( X3
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X3 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_3515_aset_I3_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X3
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X3 = T )
             => ( ( plus_plus_int @ X3 @ D4 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_3516_aset_I4_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A4 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X3
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X3 != T )
             => ( ( plus_plus_int @ X3 @ D4 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_3517_aset_I5_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A4 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X3
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X3 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_3518_Bolzano,axiom,
    ! [A: real,B: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A3: real,B3: real,C2: real] :
            ( ( P @ A3 @ B3 )
           => ( ( P @ B3 @ C2 )
             => ( ( ord_less_eq_real @ A3 @ B3 )
               => ( ( ord_less_eq_real @ B3 @ C2 )
                 => ( P @ A3 @ C2 ) ) ) ) )
       => ( ! [X5: real] :
              ( ( ord_less_eq_real @ A @ X5 )
             => ( ( ord_less_eq_real @ X5 @ B )
               => ? [D5: real] :
                    ( ( ord_less_real @ zero_zero_real @ D5 )
                    & ! [A3: real,B3: real] :
                        ( ( ( ord_less_eq_real @ A3 @ X5 )
                          & ( ord_less_eq_real @ X5 @ B3 )
                          & ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D5 ) )
                       => ( P @ A3 @ B3 ) ) ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_3519_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_3520_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3521_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_3522_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_3523_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3524_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_3525_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_3526_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_3527_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_3528_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3529_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > complex,Y: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3530_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3531_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3532_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3533_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3534_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3535_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3536_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > rat,Y: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3537_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > rat,Y: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3538_eucl__rel__int__dividesI,axiom,
    ! [L: int,K2: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K2
          = ( times_times_int @ Q3 @ L ) )
       => ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q3 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_3539_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( ( K2
          = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R2 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
            & ( ord_less_int @ R2 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R2 )
                & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q3 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_3540_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_3541_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3542_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_3543_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_3544_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3545_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > complex,Y: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3546_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3547_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3548_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3549_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3550_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I6 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3551_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I3: complex] :
              ( ( member_complex @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I3: complex] :
                ( ( member_complex @ I3 @ I6 )
                & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3552_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > rat,Y: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I6 )
                & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3553_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > rat,Y: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I3: real] :
              ( ( member_real @ I3 @ I6 )
              & ( ( X @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( Y @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I3: real] :
                ( ( member_real @ I3 @ I6 )
                & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3554_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M2: nat,N: nat] :
      ( ! [M: nat] : ( P @ M @ zero_zero_nat )
     => ( ! [M: nat,N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ( P @ N2 @ ( modulo_modulo_nat @ M @ N2 ) )
             => ( P @ M @ N2 ) ) )
       => ( P @ M2 @ N ) ) ) ).

% gcd_nat_induct
thf(fact_3555_concat__bit__Suc,axiom,
    ! [N: nat,K2: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N ) @ K2 @ L )
      = ( plus_plus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_3556_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3557_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3558_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3559_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3560_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3561_even__succ__mod__exp,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3562_even__succ__mod__exp,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3563_option_Osize__gen_I2_J,axiom,
    ! [X: product_prod_nat_nat > nat,X2: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_3564_option_Osize__gen_I2_J,axiom,
    ! [X: nat > nat,X2: nat] :
      ( ( size_option_nat @ X @ ( some_nat @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_3565_option_Osize__gen_I2_J,axiom,
    ! [X: num > nat,X2: num] :
      ( ( size_option_num @ X @ ( some_num @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_3566_nat__dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( ( dvd_dvd_nat @ M2 @ one_one_nat )
      = ( M2 = one_one_nat ) ) ).

% nat_dvd_1_iff_1
thf(fact_3567_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
thf(fact_3568_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
      = ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
thf(fact_3569_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_3570_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_3571_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_3572_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_3573_dvd__0__right,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).

% dvd_0_right
thf(fact_3574_dvd__0__right,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).

% dvd_0_right
thf(fact_3575_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_3576_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_3577_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_3578_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_3579_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3580_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_3581_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3582_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_3583_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_3584_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3585_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_3586_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3587_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_3588_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_3589_dvd__1__left,axiom,
    ! [K2: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K2 ) ).

% dvd_1_left
thf(fact_3590_dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( ( dvd_dvd_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = ( M2
        = ( suc @ zero_zero_nat ) ) ) ).

% dvd_1_iff_1
thf(fact_3591_div__dvd__div,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_3592_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_3593_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_3594_nat__mult__dvd__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
      = ( ( K2 = zero_zero_nat )
        | ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_3595_concat__bit__0,axiom,
    ! [K2: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K2 @ L )
      = L ) ).

% concat_bit_0
thf(fact_3596_dbl__simps_I2_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% dbl_simps(2)
thf(fact_3597_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_3598_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_3599_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_3600_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3601_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_3602_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_3603_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3604_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_3605_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3606_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_3607_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_3608_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3609_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_3610_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_3611_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_3612_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_3613_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_3614_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_3615_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_3616_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_prod
thf(fact_3617_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_3618_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_3619_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3620_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_3621_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3622_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_3623_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_3624_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3625_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_3626_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3627_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_3628_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_3629_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_3630_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_3631_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_3632_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_3633_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_3634_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_3635_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_div
thf(fact_3636_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_3637_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_3638_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).

% unit_div_1_unit
thf(fact_3639_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_3640_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_3641_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_3642_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_3643_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_3644_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3645_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_3646_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_3647_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
          = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).

% div_diff
thf(fact_3648_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_3649_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( modulo364778990260209775nteger @ B @ A )
        = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
thf(fact_3650_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_3651_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_3652_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K2 @ L ) )
      = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_3653_concat__bit__negative__iff,axiom,
    ! [N: nat,K2: int,L: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N @ K2 @ L ) @ zero_zero_int )
      = ( ord_less_int @ L @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_3654_dbl__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K2 ) )
      = ( numera6690914467698888265omplex @ ( bit0 @ K2 ) ) ) ).

% dbl_simps(5)
thf(fact_3655_dbl__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K2 ) )
      = ( numeral_numeral_real @ ( bit0 @ K2 ) ) ) ).

% dbl_simps(5)
thf(fact_3656_dbl__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K2 ) )
      = ( numeral_numeral_rat @ ( bit0 @ K2 ) ) ) ).

% dbl_simps(5)
thf(fact_3657_dbl__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) ).

% dbl_simps(5)
thf(fact_3658_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = ( divide6298287555418463151nteger @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_3659_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_3660_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_3661_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_3662_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_3663_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_3664_even__Suc,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% even_Suc
thf(fact_3665_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_Suc_Suc_iff
thf(fact_3666_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_3667_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_3668_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_3669_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_3670_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_3671_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_3672_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_3673_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_3674_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_3675_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_3676_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_3677_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_3678_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_3679_even__mod__2__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_3680_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_3681_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_3682_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_3683_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_3684_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_3685_power__less__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_3686_power__less__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_3687_power__less__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_3688_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_3689_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_3690_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_3691_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_3692_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_3693_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_3694_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).

% even_diff
thf(fact_3695_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_3696_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% odd_Suc_minus_one
thf(fact_3697_even__diff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ M2 @ N )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).

% even_diff_nat
thf(fact_3698_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_3699_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_3700_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_3701_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_3702_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_3703_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_3704_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% odd_succ_div_two
thf(fact_3705_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_3706_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_3707_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_3708_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_3709_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_3710_even__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_3711_even__power,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_3712_even__power,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_3713_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_3714_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_3715_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_3716_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_3717_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_3718_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_3719_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_3720_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3721_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3722_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3723_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3724_even__succ__div__exp,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3725_even__succ__div__exp,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3726_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_3727_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_3728_dvd__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ B @ C )
       => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_trans
thf(fact_3729_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_3730_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_3731_dvd__refl,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).

% dvd_refl
thf(fact_3732_dvd__productE,axiom,
    ! [P2: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A @ B ) )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( P2
              = ( times_times_nat @ X5 @ Y4 ) )
           => ( ( dvd_dvd_nat @ X5 @ A )
             => ~ ( dvd_dvd_nat @ Y4 @ B ) ) ) ) ).

% dvd_productE
thf(fact_3733_dvd__productE,axiom,
    ! [P2: int,A: int,B: int] :
      ( ( dvd_dvd_int @ P2 @ ( times_times_int @ A @ B ) )
     => ~ ! [X5: int,Y4: int] :
            ( ( P2
              = ( times_times_int @ X5 @ Y4 ) )
           => ( ( dvd_dvd_int @ X5 @ A )
             => ~ ( dvd_dvd_int @ Y4 @ B ) ) ) ) ).

% dvd_productE
thf(fact_3734_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
     => ? [B7: nat,C5: nat] :
          ( ( A
            = ( times_times_nat @ B7 @ C5 ) )
          & ( dvd_dvd_nat @ B7 @ B )
          & ( dvd_dvd_nat @ C5 @ C ) ) ) ).

% division_decomp
thf(fact_3735_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
     => ? [B7: int,C5: int] :
          ( ( A
            = ( times_times_int @ B7 @ C5 ) )
          & ( dvd_dvd_int @ B7 @ B )
          & ( dvd_dvd_int @ C5 @ C ) ) ) ).

% division_decomp
thf(fact_3736_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
thf(fact_3737_dvd__0__left,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
thf(fact_3738_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_3739_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_3740_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_3741_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_3742_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ~ ! [K: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ B @ K ) ) ) ).

% dvdE
thf(fact_3743_dvdE,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ~ ! [K: real] :
            ( A
           != ( times_times_real @ B @ K ) ) ) ).

% dvdE
thf(fact_3744_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ~ ! [K: rat] :
            ( A
           != ( times_times_rat @ B @ K ) ) ) ).

% dvdE
thf(fact_3745_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K: nat] :
            ( A
           != ( times_times_nat @ B @ K ) ) ) ).

% dvdE
thf(fact_3746_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K: int] :
            ( A
           != ( times_times_int @ B @ K ) ) ) ).

% dvdE
thf(fact_3747_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K2: code_integer] :
      ( ( A
        = ( times_3573771949741848930nteger @ B @ K2 ) )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% dvdI
thf(fact_3748_dvdI,axiom,
    ! [A: real,B: real,K2: real] :
      ( ( A
        = ( times_times_real @ B @ K2 ) )
     => ( dvd_dvd_real @ B @ A ) ) ).

% dvdI
thf(fact_3749_dvdI,axiom,
    ! [A: rat,B: rat,K2: rat] :
      ( ( A
        = ( times_times_rat @ B @ K2 ) )
     => ( dvd_dvd_rat @ B @ A ) ) ).

% dvdI
thf(fact_3750_dvdI,axiom,
    ! [A: nat,B: nat,K2: nat] :
      ( ( A
        = ( times_times_nat @ B @ K2 ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_3751_dvdI,axiom,
    ! [A: int,B: int,K2: int] :
      ( ( A
        = ( times_times_int @ B @ K2 ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_3752_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B5: code_integer,A5: code_integer] :
        ? [K3: code_integer] :
          ( A5
          = ( times_3573771949741848930nteger @ B5 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_3753_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B5: real,A5: real] :
        ? [K3: real] :
          ( A5
          = ( times_times_real @ B5 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_3754_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B5: rat,A5: rat] :
        ? [K3: rat] :
          ( A5
          = ( times_times_rat @ B5 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_3755_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B5: nat,A5: nat] :
        ? [K3: nat] :
          ( A5
          = ( times_times_nat @ B5 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_3756_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B5: int,A5: int] :
        ? [K3: int] :
          ( A5
          = ( times_times_int @ B5 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_3757_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3758_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_3759_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3760_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_3761_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_3762_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3763_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_3764_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3765_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_3766_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_3767_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3768_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_3769_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3770_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_3771_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_3772_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3773_dvd__triv__left,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3774_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3775_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3776_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3777_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ C @ D )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3778_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_3779_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ C @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3780_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_3781_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_3782_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3783_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_3784_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3785_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_3786_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_3787_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3788_dvd__triv__right,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3789_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3790_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3791_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3792_one__dvd,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).

% one_dvd
thf(fact_3793_one__dvd,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).

% one_dvd
thf(fact_3794_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_3795_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_3796_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_3797_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_3798_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_3799_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_3800_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_3801_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).

% dvd_unit_imp_unit
thf(fact_3802_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_3803_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_3804_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3805_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_3806_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ C )
       => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3807_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_3808_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_3809_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3810_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_3811_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3812_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_3813_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_3814_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3815_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_3816_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3817_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_3818_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_3819_dvd__diff,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( ( dvd_dvd_Code_integer @ X @ Z )
       => ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_3820_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_3821_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( dvd_dvd_rat @ X @ Y )
     => ( ( dvd_dvd_rat @ X @ Z )
       => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_3822_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_3823_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
      = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).

% dvd_diff_commute
thf(fact_3824_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).

% dvd_diff_commute
thf(fact_3825_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( ( divide6298287555418463151nteger @ A @ C )
            = ( divide6298287555418463151nteger @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3826_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( dvd_dvd_complex @ C @ A )
     => ( ( dvd_dvd_complex @ C @ B )
       => ( ( ( divide1717551699836669952omplex @ A @ C )
            = ( divide1717551699836669952omplex @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3827_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_3828_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ C @ A )
     => ( ( dvd_dvd_rat @ C @ B )
       => ( ( ( divide_divide_rat @ A @ C )
            = ( divide_divide_rat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3829_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_3830_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_3831_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ C )
        = ( divide6298287555418463151nteger @ B @ C ) )
     => ( ( dvd_dvd_Code_integer @ C @ A )
       => ( ( dvd_dvd_Code_integer @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3832_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B @ C ) )
     => ( ( dvd_dvd_complex @ C @ A )
       => ( ( dvd_dvd_complex @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3833_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_3834_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
     => ( ( dvd_dvd_rat @ C @ A )
       => ( ( dvd_dvd_rat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3835_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_3836_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_3837_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ B )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_3838_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_3839_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_3840_dvd__power__same,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_3841_dvd__power__same,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_3842_dvd__power__same,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_3843_dvd__power__same,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_3844_dvd__power__same,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_3845_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
        = ( modulo364778990260209775nteger @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_3846_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
        = ( modulo_modulo_nat @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_3847_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
        = ( modulo_modulo_int @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_3848_dvd__mod,axiom,
    ! [K2: code_integer,M2: code_integer,N: code_integer] :
      ( ( dvd_dvd_Code_integer @ K2 @ M2 )
     => ( ( dvd_dvd_Code_integer @ K2 @ N )
       => ( dvd_dvd_Code_integer @ K2 @ ( modulo364778990260209775nteger @ M2 @ N ) ) ) ) ).

% dvd_mod
thf(fact_3849_dvd__mod,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ K2 @ M2 )
     => ( ( dvd_dvd_nat @ K2 @ N )
       => ( dvd_dvd_nat @ K2 @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ).

% dvd_mod
thf(fact_3850_dvd__mod,axiom,
    ! [K2: int,M2: int,N: int] :
      ( ( dvd_dvd_int @ K2 @ M2 )
     => ( ( dvd_dvd_int @ K2 @ N )
       => ( dvd_dvd_int @ K2 @ ( modulo_modulo_int @ M2 @ N ) ) ) ) ).

% dvd_mod
thf(fact_3851_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_3852_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_3853_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_3854_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
        = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_3855_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_3856_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_3857_dvd__diff__nat,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ K2 @ M2 )
     => ( ( dvd_dvd_nat @ K2 @ N )
       => ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% dvd_diff_nat
thf(fact_3858_zdvd__zdiffD,axiom,
    ! [K2: int,M2: int,N: int] :
      ( ( dvd_dvd_int @ K2 @ ( minus_minus_int @ M2 @ N ) )
     => ( ( dvd_dvd_int @ K2 @ N )
       => ( dvd_dvd_int @ K2 @ M2 ) ) ) ).

% zdvd_zdiffD
thf(fact_3859_dvd__pos__nat,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ M2 @ N )
       => ( ord_less_nat @ zero_zero_nat @ M2 ) ) ) ).

% dvd_pos_nat
thf(fact_3860_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X5: nat,Y4: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( times_times_nat @ A @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) )
        | ( ( times_times_nat @ B @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D3 ) ) ) ) ).

% bezout_add_nat
thf(fact_3861_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,Y4: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X5 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X5 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_3862_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X5: nat,Y4: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B @ Y4 ) )
          = D3 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
          = D3 ) ) ) ).

% bezout1_nat
thf(fact_3863_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_3864_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_set_real
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_3865_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_3866_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le7084787975880047091nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_3867_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_3868_concat__bit__assoc,axiom,
    ! [N: nat,K2: int,M2: nat,L: int,R2: int] :
      ( ( bit_concat_bit @ N @ K2 @ ( bit_concat_bit @ M2 @ L @ R2 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M2 @ N ) @ ( bit_concat_bit @ N @ K2 @ L ) @ R2 ) ) ).

% concat_bit_assoc
thf(fact_3869_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_set_complex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
      = ( ( dvd_dvd_complex @ A @ B )
        & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_3870_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_set_real
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
      = ( ( dvd_dvd_real @ A @ B )
        & ~ ( dvd_dvd_real @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_3871_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_3872_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( ( dvd_dvd_int @ A @ B )
        & ~ ( dvd_dvd_int @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_3873_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le1307284697595431911nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
      = ( ( dvd_dvd_Code_integer @ A @ B )
        & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_3874_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_3875_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_3876_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_3877_pinf_I9_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X3 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S2 ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_3878_pinf_I9_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z3 @ X3 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S2 ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_3879_pinf_I9_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z3 @ X3 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S2 ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_3880_pinf_I9_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S2 ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_3881_pinf_I9_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z3 @ X3 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S2 ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_3882_pinf_I10_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X3 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S2 ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_3883_pinf_I10_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z3 @ X3 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S2 ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_3884_pinf_I10_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z3 @ X3 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S2 ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_3885_pinf_I10_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z3 @ X3 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S2 ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_3886_pinf_I10_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z3 @ X3 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S2 ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_3887_minf_I9_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X3 @ Z3 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S2 ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_3888_minf_I9_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z3 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S2 ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_3889_minf_I9_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z3 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S2 ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_3890_minf_I9_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S2 ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_3891_minf_I9_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z3 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S2 ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_3892_minf_I10_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X3 @ Z3 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S2 ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_3893_minf_I10_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z3 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S2 ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_3894_minf_I10_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z3 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S2 ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_3895_minf_I10_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z3 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S2 ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_3896_minf_I10_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z3 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S2 ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_3897_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3898_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_3899_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_3900_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( ( divide_divide_rat @ A @ B )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3901_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_3902_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_3903_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).

% is_unit_mult_iff
thf(fact_3904_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_3905_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_3906_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_3907_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_3908_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_3909_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_3910_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_3911_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_3912_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_3913_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_3914_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_3915_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_3916_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_3917_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_3918_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ A @ B )
          = ( times_3573771949741848930nteger @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_3919_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_3920_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_3921_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ B @ A )
          = ( times_3573771949741848930nteger @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_3922_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_3923_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_3924_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_3925_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_3926_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_3927_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_3928_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_3929_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_3930_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_3931_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_3932_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_3933_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_3934_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_3935_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_3936_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
     => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_3937_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_3938_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_3939_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( dvd_dvd_Code_integer @ D @ C )
       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_3940_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_3941_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_3942_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ B @ A )
          = ( divide6298287555418463151nteger @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_3943_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_3944_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_3945_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_3946_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_3947_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_3948_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_3949_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_3950_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_3951_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3952_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_3953_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_3954_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3955_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_3956_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_3957_div__power,axiom,
    ! [B: code_integer,A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
        = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).

% div_power
thf(fact_3958_div__power,axiom,
    ! [B: nat,A: nat,N: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
        = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).

% div_power
thf(fact_3959_div__power,axiom,
    ! [B: int,A: int,N: nat] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
        = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% div_power
thf(fact_3960_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_3961_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_3962_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_3963_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A5: code_integer,B5: code_integer] :
          ( ( modulo364778990260209775nteger @ B5 @ A5 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_3964_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A5: nat,B5: nat] :
          ( ( modulo_modulo_nat @ B5 @ A5 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_3965_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A5: int,B5: int] :
          ( ( modulo_modulo_int @ B5 @ A5 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_3966_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_3967_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_3968_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_3969_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_3970_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_3971_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_3972_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_3973_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: complex] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( dvd_dvd_complex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_3974_power__le__dvd,axiom,
    ! [A: code_integer,N: nat,B: code_integer,M2: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M2 ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3975_power__le__dvd,axiom,
    ! [A: nat,N: nat,B: nat,M2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M2 ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3976_power__le__dvd,axiom,
    ! [A: real,N: nat,B: real,M2: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M2 ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3977_power__le__dvd,axiom,
    ! [A: int,N: nat,B: int,M2: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M2 ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3978_power__le__dvd,axiom,
    ! [A: complex,N: nat,B: complex,M2: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M2 ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3979_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,N: nat,M2: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_3980_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N: nat,M2: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_3981_dvd__power__le,axiom,
    ! [X: real,Y: real,N: nat,M2: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_3982_dvd__power__le,axiom,
    ! [X: int,Y: int,N: nat,M2: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_3983_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N: nat,M2: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_3984_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
      = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% mod_eq_dvd_iff
thf(fact_3985_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
      = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).

% mod_eq_dvd_iff
thf(fact_3986_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_3987_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_3988_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_3989_nat__dvd__not__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).

% nat_dvd_not_less
thf(fact_3990_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X5: nat,Y4: nat] :
          ( ( dvd_dvd_nat @ D3 @ A )
          & ( dvd_dvd_nat @ D3 @ B )
          & ( ( times_times_nat @ A @ X5 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_3991_dvd__minus__self,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% dvd_minus_self
thf(fact_3992_less__eq__dvd__minus,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( dvd_dvd_nat @ M2 @ N )
        = ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_3993_dvd__diffD1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
     => ( ( dvd_dvd_nat @ K2 @ M2 )
       => ( ( ord_less_eq_nat @ N @ M2 )
         => ( dvd_dvd_nat @ K2 @ N ) ) ) ) ).

% dvd_diffD1
thf(fact_3994_dvd__diffD,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
     => ( ( dvd_dvd_nat @ K2 @ N )
       => ( ( ord_less_eq_nat @ N @ M2 )
         => ( dvd_dvd_nat @ K2 @ M2 ) ) ) ) ).

% dvd_diffD
thf(fact_3995_zdvd__mono,axiom,
    ! [K2: int,M2: int,T: int] :
      ( ( K2 != zero_zero_int )
     => ( ( dvd_dvd_int @ M2 @ T )
        = ( dvd_dvd_int @ ( times_times_int @ K2 @ M2 ) @ ( times_times_int @ K2 @ T ) ) ) ) ).

% zdvd_mono
thf(fact_3996_zdvd__mult__cancel,axiom,
    ! [K2: int,M2: int,N: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K2 @ M2 ) @ ( times_times_int @ K2 @ N ) )
     => ( ( K2 != zero_zero_int )
       => ( dvd_dvd_int @ M2 @ N ) ) ) ).

% zdvd_mult_cancel
thf(fact_3997_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X4: real] : ( plus_plus_real @ X4 @ X4 ) ) ) ).

% dbl_def
thf(fact_3998_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X4: rat] : ( plus_plus_rat @ X4 @ X4 ) ) ) ).

% dbl_def
thf(fact_3999_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X4: int] : ( plus_plus_int @ X4 @ X4 ) ) ) ).

% dbl_def
thf(fact_4000_zdvd__reduce,axiom,
    ! [K2: int,N: int,M2: int] :
      ( ( dvd_dvd_int @ K2 @ ( plus_plus_int @ N @ ( times_times_int @ K2 @ M2 ) ) )
      = ( dvd_dvd_int @ K2 @ N ) ) ).

% zdvd_reduce
thf(fact_4001_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_4002_finite__divisors__nat,axiom,
    ! [M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M2 ) ) ) ) ).

% finite_divisors_nat
thf(fact_4003_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] :
              ( B
             != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_4004_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] :
              ( B
             != ( times_times_nat @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_4005_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] :
              ( B
             != ( times_times_int @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_4006_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L: code_integer] :
      ( ( ? [X4: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X4 ) ) )
      = ( ? [X4: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X4 @ zero_z3403309356797280102nteger ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4007_unity__coeff__ex,axiom,
    ! [P: complex > $o,L: complex] :
      ( ( ? [X4: complex] : ( P @ ( times_times_complex @ L @ X4 ) ) )
      = ( ? [X4: complex] :
            ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X4 @ zero_zero_complex ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4008_unity__coeff__ex,axiom,
    ! [P: real > $o,L: real] :
      ( ( ? [X4: real] : ( P @ ( times_times_real @ L @ X4 ) ) )
      = ( ? [X4: real] :
            ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X4 @ zero_zero_real ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4009_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X4: rat] : ( P @ ( times_times_rat @ L @ X4 ) ) )
      = ( ? [X4: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X4 @ zero_zero_rat ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4010_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X4: nat] : ( P @ ( times_times_nat @ L @ X4 ) ) )
      = ( ? [X4: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X4 @ zero_zero_nat ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4011_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X4: int] : ( P @ ( times_times_int @ L @ X4 ) ) )
      = ( ? [X4: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X4 @ zero_zero_int ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4012_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ A @ B )
       => ( ( ( divide6298287555418463151nteger @ B @ A )
            = C )
          = ( B
            = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4013_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_4014_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_4015_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
          = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4016_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_4017_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_4018_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4019_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_4020_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_4021_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( ( dvd_dvd_Code_integer @ A @ B )
         => ( ( dvd_dvd_Code_integer @ C @ D )
           => ( ( ( divide6298287555418463151nteger @ B @ A )
                = ( divide6298287555418463151nteger @ D @ C ) )
              = ( ( times_3573771949741848930nteger @ B @ C )
                = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4022_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_4023_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_4024_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
thf(fact_4025_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_4026_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_4027_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_4028_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_4029_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_4030_inf__period_I4_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X3: code_integer,K4: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ T ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4031_inf__period_I4_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [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 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4032_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X3: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4033_inf__period_I4_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [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 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4034_inf__period_I3_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X3: code_integer,K4: code_integer] :
          ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ T ) )
          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4035_inf__period_I3_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [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 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4036_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X3: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ T ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4037_inf__period_I3_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [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 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4038_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = C )
        = ( A
          = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_4039_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_4040_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_4041_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( A
          = ( divide6298287555418463151nteger @ C @ B ) )
        = ( ( times_3573771949741848930nteger @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_4042_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_4043_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_4044_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_4045_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_4046_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_4047_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_4048_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_4049_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_4050_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_4051_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_4052_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_4053_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_4054_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_4055_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_4056_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_4057_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_4058_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_4059_is__unit__power__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4060_is__unit__power__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4061_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4062_dvd__imp__le,axiom,
    ! [K2: nat,N: nat] :
      ( ( dvd_dvd_nat @ K2 @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ K2 @ N ) ) ) ).

% dvd_imp_le
thf(fact_4063_nat__mult__dvd__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
        = ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_4064_dvd__mult__cancel,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% dvd_mult_cancel
thf(fact_4065_mod__greater__zero__iff__not__dvd,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ N ) )
      = ( ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_4066_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M2: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( ( modulo_modulo_nat @ M2 @ Q3 )
          = ( modulo_modulo_nat @ N @ Q3 ) )
        = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_4067_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_4068_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_4069_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_4070_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B3: code_integer] :
              ( ( B3 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B3 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B3 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4071_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B3 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B3 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B3 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4072_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( ( dvd_dvd_int @ B3 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B3 )
                 => ( ( ( divide_divide_int @ one_one_int @ B3 )
                      = A )
                   => ( ( ( times_times_int @ A @ B3 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4073_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4074_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_4075_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_4076_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4077_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_4078_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_4079_evenE,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% evenE
thf(fact_4080_evenE,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: nat] :
            ( A
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% evenE
thf(fact_4081_evenE,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: int] :
            ( A
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% evenE
thf(fact_4082_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_4083_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_4084_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_4085_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_4086_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_4087_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_4088_bit__eq__rec,axiom,
    ( ( ^ [Y6: code_integer,Z4: code_integer] : Y6 = Z4 )
    = ( ^ [A5: code_integer,B5: code_integer] :
          ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 )
            = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) )
          & ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = ( divide6298287555418463151nteger @ B5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4089_bit__eq__rec,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [A5: nat,B5: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) )
          & ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4090_bit__eq__rec,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [A5: int,B5: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) )
          & ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4091_dvd__power__iff,axiom,
    ! [X: code_integer,M2: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M2 ) @ ( power_8256067586552552935nteger @ X @ N ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4092_dvd__power__iff,axiom,
    ! [X: nat,M2: nat,N: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M2 ) @ ( power_power_nat @ X @ N ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4093_dvd__power__iff,axiom,
    ! [X: int,M2: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M2 ) @ ( power_power_int @ X @ N ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4094_dvd__power,axiom,
    ! [N: nat,X: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).

% dvd_power
thf(fact_4095_dvd__power,axiom,
    ! [N: nat,X: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_rat ) )
     => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).

% dvd_power
thf(fact_4096_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).

% dvd_power
thf(fact_4097_dvd__power,axiom,
    ! [N: nat,X: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_real ) )
     => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).

% dvd_power
thf(fact_4098_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).

% dvd_power
thf(fact_4099_dvd__power,axiom,
    ! [N: nat,X: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_complex ) )
     => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).

% dvd_power
thf(fact_4100_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
thf(fact_4101_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_4102_dvd__mult__cancel2,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M2 ) @ M2 )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_4103_dvd__mult__cancel1,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M2 @ N ) @ M2 )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_4104_dvd__minus__add,axiom,
    ! [Q3: nat,N: nat,R2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ Q3 @ N )
     => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R2 @ M2 ) )
       => ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ Q3 ) )
          = ( dvd_dvd_nat @ M2 @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M2 ) @ Q3 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_4105_power__dvd__imp__le,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_4106_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ R2 @ N )
     => ( ( ord_less_eq_nat @ R2 @ M2 )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M2 @ R2 ) )
         => ( ( modulo_modulo_nat @ M2 @ N )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_4107_aset_I10_J,axiom,
    ! [D: int,D4: int,A4: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A4 )
                 => ( X3
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).

% aset(10)
thf(fact_4108_aset_I9_J,axiom,
    ! [D: int,D4: int,A4: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A4 )
                 => ( X3
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).

% aset(9)
thf(fact_4109_bset_I10_J,axiom,
    ! [D: int,D4: int,B4: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B4 )
                 => ( X3
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).

% bset(10)
thf(fact_4110_bset_I9_J,axiom,
    ! [D: int,D4: int,B4: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B4 )
                 => ( X3
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).

% bset(9)
thf(fact_4111_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_4112_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_4113_even__two__times__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_4114_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_4115_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_4116_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_4117_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4118_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4119_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4120_power__mono__odd,axiom,
    ! [N: nat,A: real,B: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_4121_power__mono__odd,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_4122_power__mono__odd,axiom,
    ! [N: nat,A: int,B: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_4123_odd__pos,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% odd_pos
thf(fact_4124_dvd__power__iff__le,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K2 @ M2 ) @ ( power_power_nat @ K2 @ N ) )
        = ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% dvd_power_iff_le
thf(fact_4125_even__unset__bit__iff,axiom,
    ! [M2: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M2 @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_4126_even__unset__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M2 @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_4127_even__unset__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M2 @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_4128_even__set__bit__iff,axiom,
    ! [M2: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M2 @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_4129_even__set__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M2 @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_4130_even__set__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M2 @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_4131_even__flip__bit__iff,axiom,
    ! [M2: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M2 @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       != ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_4132_even__flip__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M2 @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       != ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_4133_even__flip__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M2 @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       != ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_4134_even__diff__iff,axiom,
    ! [K2: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K2 @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).

% even_diff_iff
thf(fact_4135_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_4136_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_4137_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_4138_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = one_one_Code_integer ) ) ) ).

% mod2_eq_if
thf(fact_4139_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_4140_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_4141_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) )
     => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
           != one_one_Code_integer ) ) ) ).

% parity_cases
thf(fact_4142_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_4143_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_4144_zero__le__power__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_4145_zero__le__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_4146_zero__le__power__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_4147_zero__le__odd__power,axiom,
    ! [N: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).

% zero_le_odd_power
thf(fact_4148_zero__le__odd__power,axiom,
    ! [N: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).

% zero_le_odd_power
thf(fact_4149_zero__le__odd__power,axiom,
    ! [N: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_odd_power
thf(fact_4150_zero__le__even__power,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_4151_zero__le__even__power,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_4152_zero__le__even__power,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_4153_zero__less__power__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_4154_zero__less__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_4155_zero__less__power__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_4156_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P @ A3 @ B3 )
          = ( P @ B3 @ A3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ A3 @ B3 )
             => ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_4157_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_4158_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_4159_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_4160_power__le__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_4161_power__le__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_4162_power__le__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_4163_option_Osize__gen_I1_J,axiom,
    ! [X: nat > nat] :
      ( ( size_option_nat @ X @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_4164_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_4165_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_4166_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_4167_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4168_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4169_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4170_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4171_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4172_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4173_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ S3 )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S3 )
                  & ( ord_less_real @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4174_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( ( finite_finite_rat @ S3 )
     => ( ( S3 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ S3 )
            & ~ ? [Xa: rat] :
                  ( ( member_rat @ Xa @ S3 )
                  & ( ord_less_rat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4175_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( ( finite_finite_num @ S3 )
     => ( ( S3 != bot_bot_set_num )
       => ? [X5: num] :
            ( ( member_num @ X5 @ S3 )
            & ~ ? [Xa: num] :
                  ( ( member_num @ Xa @ S3 )
                  & ( ord_less_num @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4176_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ S3 )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S3 )
                  & ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4177_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ S3 )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S3 )
                  & ( ord_less_int @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4178_infinite__growing,axiom,
    ! [X9: set_real] :
      ( ( X9 != bot_bot_set_real )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ X9 )
           => ? [Xa: real] :
                ( ( member_real @ Xa @ X9 )
                & ( ord_less_real @ X5 @ Xa ) ) )
       => ~ ( finite_finite_real @ X9 ) ) ) ).

% infinite_growing
thf(fact_4179_infinite__growing,axiom,
    ! [X9: set_rat] :
      ( ( X9 != bot_bot_set_rat )
     => ( ! [X5: rat] :
            ( ( member_rat @ X5 @ X9 )
           => ? [Xa: rat] :
                ( ( member_rat @ Xa @ X9 )
                & ( ord_less_rat @ X5 @ Xa ) ) )
       => ~ ( finite_finite_rat @ X9 ) ) ) ).

% infinite_growing
thf(fact_4180_infinite__growing,axiom,
    ! [X9: set_num] :
      ( ( X9 != bot_bot_set_num )
     => ( ! [X5: num] :
            ( ( member_num @ X5 @ X9 )
           => ? [Xa: num] :
                ( ( member_num @ Xa @ X9 )
                & ( ord_less_num @ X5 @ Xa ) ) )
       => ~ ( finite_finite_num @ X9 ) ) ) ).

% infinite_growing
thf(fact_4181_infinite__growing,axiom,
    ! [X9: set_nat] :
      ( ( X9 != bot_bot_set_nat )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ X9 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X9 )
                & ( ord_less_nat @ X5 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X9 ) ) ) ).

% infinite_growing
thf(fact_4182_infinite__growing,axiom,
    ! [X9: set_int] :
      ( ( X9 != bot_bot_set_int )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ X9 )
           => ? [Xa: int] :
                ( ( member_int @ Xa @ X9 )
                & ( ord_less_int @ X5 @ Xa ) ) )
       => ~ ( finite_finite_int @ X9 ) ) ) ).

% infinite_growing
thf(fact_4183_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 ) ) )
         => ~ ! [Va3: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_4184_flip__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
      = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4185_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_4186_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_4187_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_4188_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
      = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_4189_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_4190_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_4191_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_4192_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_4193_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_4194_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4195_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_4196_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_4197_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X4: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( minus_minus_real @ one_one_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_4198_Sum__Icc__int,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_eq_int @ M2 @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X4: int] : X4
          @ ( set_or1266510415728281911st_int @ M2 @ N ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M2 @ ( minus_minus_int @ M2 @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_4199_intind,axiom,
    ! [I: nat,N: nat,P: nat > $o,X: nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X )
       => ( P @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I ) ) ) ) ).

% intind
thf(fact_4200_intind,axiom,
    ! [I: nat,N: nat,P: int > $o,X: int] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X )
       => ( P @ ( nth_int @ ( replicate_int @ N @ X ) @ I ) ) ) ) ).

% intind
thf(fact_4201_intind,axiom,
    ! [I: nat,N: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X )
       => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I ) ) ) ) ).

% intind
thf(fact_4202_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_4203_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_4204_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_4205_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_4206_of__bool__eq_I1_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $false )
    = zero_zero_complex ) ).

% of_bool_eq(1)
thf(fact_4207_of__bool__eq_I1_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $false )
    = zero_zero_real ) ).

% of_bool_eq(1)
thf(fact_4208_of__bool__eq_I1_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $false )
    = zero_zero_rat ) ).

% of_bool_eq(1)
thf(fact_4209_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_4210_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_4211_of__bool__eq_I1_J,axiom,
    ( ( zero_n356916108424825756nteger @ $false )
    = zero_z3403309356797280102nteger ) ).

% of_bool_eq(1)
thf(fact_4212_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = zero_zero_complex )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4213_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = zero_zero_real )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4214_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = zero_zero_rat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4215_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = zero_zero_nat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4216_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = zero_zero_int )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4217_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = zero_z3403309356797280102nteger )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4218_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4219_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4220_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4221_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4222_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4223_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = one_one_complex )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4224_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = one_one_real )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4225_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = one_one_rat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4226_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4227_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4228_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = one_one_Code_integer )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4229_of__bool__eq_I2_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $true )
    = one_one_complex ) ).

% of_bool_eq(2)
thf(fact_4230_of__bool__eq_I2_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $true )
    = one_one_real ) ).

% of_bool_eq(2)
thf(fact_4231_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_4232_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_4233_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_4234_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_4235_signed__take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% signed_take_bit_of_0
thf(fact_4236_replicate__eq__replicate,axiom,
    ! [M2: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M2 @ X )
        = ( replicate_VEBT_VEBT @ N @ Y ) )
      = ( ( M2 = N )
        & ( ( M2 != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_4237_length__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_4238_length__replicate,axiom,
    ! [N: nat,X: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_4239_length__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_4240_length__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( size_size_list_int @ ( replicate_int @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_4241_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P
          | Q ) )
      = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_4242_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P
          | Q ) )
      = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_4243_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P
          | Q ) )
      = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_4244_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4245_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4246_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4247_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4248_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4249_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_4250_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
        = ( ord_less_real @ one_one_real @ X ) ) ) ).

% ln_gt_zero_iff
thf(fact_4251_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ln_ln_real @ X )
          = zero_zero_real )
        = ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_4252_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4253_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4254_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4255_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4256_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4257_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_4258_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex @ ~ P )
      = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4259_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real @ ~ P )
      = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4260_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat @ ~ P )
      = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4261_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2684676970156552555ol_int @ ~ P )
      = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4262_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger @ ~ P )
      = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4263_Suc__0__mod__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( zero_n2687167440665602831ol_nat
        @ ( N
         != ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_4264_signed__take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_4265_signed__take__bit__numeral__of__1,axiom,
    ! [K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K2 ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_numeral_of_1
thf(fact_4266_in__set__replicate,axiom,
    ! [X: real,N: nat,Y: real] :
      ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4267_in__set__replicate,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4268_in__set__replicate,axiom,
    ! [X: product_prod_nat_nat,N: nat,Y: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4269_in__set__replicate,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4270_in__set__replicate,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4271_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_4272_Bex__set__replicate,axiom,
    ! [N: nat,A: int,P: int > $o] :
      ( ( ? [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
            & ( P @ X4 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_4273_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
            & ( P @ X4 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_4274_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
            & ( P @ X4 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_4275_Ball__set__replicate,axiom,
    ! [N: nat,A: int,P: int > $o] :
      ( ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
           => ( P @ X4 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_4276_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
           => ( P @ X4 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_4277_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
           => ( P @ X4 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_4278_nth__replicate,axiom,
    ! [I: nat,N: nat,X: nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_4279_nth__replicate,axiom,
    ! [I: nat,N: nat,X: int] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_int @ ( replicate_int @ N @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_4280_nth__replicate,axiom,
    ! [I: nat,N: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_4281_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_4282_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_4283_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_iff
thf(fact_4284_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_4285_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_4286_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_4287_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_4288_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_4289_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_4290_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = zero_z3403309356797280102nteger ) ).

% of_bool_half_eq_0
thf(fact_4291_set__decode__0,axiom,
    ! [X: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).

% set_decode_0
thf(fact_4292_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4293_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4294_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4295_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4296_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4297_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4298_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4299_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4300_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4301_dvd__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ M2 @ N )
     => ( ( dvd_dvd_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% dvd_antisym
thf(fact_4302_of__bool__eq__iff,axiom,
    ! [P2: $o,Q3: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P2 )
        = ( zero_n2687167440665602831ol_nat @ Q3 ) )
      = ( P2 = Q3 ) ) ).

% of_bool_eq_iff
thf(fact_4303_of__bool__eq__iff,axiom,
    ! [P2: $o,Q3: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P2 )
        = ( zero_n2684676970156552555ol_int @ Q3 ) )
      = ( P2 = Q3 ) ) ).

% of_bool_eq_iff
thf(fact_4304_of__bool__eq__iff,axiom,
    ! [P2: $o,Q3: $o] :
      ( ( ( zero_n356916108424825756nteger @ P2 )
        = ( zero_n356916108424825756nteger @ Q3 ) )
      = ( P2 = Q3 ) ) ).

% of_bool_eq_iff
thf(fact_4305_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n3304061248610475627l_real
        @ ( P
          & Q ) )
      = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).

% of_bool_conj
thf(fact_4306_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2052037380579107095ol_rat
        @ ( P
          & Q ) )
      = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).

% of_bool_conj
thf(fact_4307_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P
          & Q ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_conj
thf(fact_4308_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P
          & Q ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_conj
thf(fact_4309_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P
          & Q ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).

% of_bool_conj
thf(fact_4310_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_4311_signed__take__bit__mult,axiom,
    ! [N: nat,K2: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K2 @ L ) ) ) ).

% signed_take_bit_mult
thf(fact_4312_signed__take__bit__add,axiom,
    ! [N: nat,K2: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K2 @ L ) ) ) ).

% signed_take_bit_add
thf(fact_4313_signed__take__bit__diff,axiom,
    ! [N: nat,K2: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K2 @ L ) ) ) ).

% signed_take_bit_diff
thf(fact_4314_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_4315_sum__mono,axiom,
    ! [K5: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_4316_sum__mono,axiom,
    ! [K5: set_int,F: int > rat,G: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_4317_sum__mono,axiom,
    ! [K5: set_real,F: real > rat,G: real > rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_4318_sum__mono,axiom,
    ! [K5: set_complex,F: complex > rat,G: complex > rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_4319_sum__mono,axiom,
    ! [K5: set_int,F: int > nat,G: int > nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_4320_sum__mono,axiom,
    ! [K5: set_real,F: real > nat,G: real > nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_4321_sum__mono,axiom,
    ! [K5: set_complex,F: complex > nat,G: complex > nat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_4322_sum__mono,axiom,
    ! [K5: set_nat,F: nat > int,G: nat > int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_4323_sum__mono,axiom,
    ! [K5: set_real,F: real > int,G: real > int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_4324_sum__mono,axiom,
    ! [K5: set_complex,F: complex > int,G: complex > int] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_4325_sum__product,axiom,
    ! [F: int > int,A4: set_int,G: int > int,B4: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ B4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I3: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J3: int] : ( times_times_int @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B4 )
        @ A4 ) ) ).

% sum_product
thf(fact_4326_sum__product,axiom,
    ! [F: complex > complex,A4: set_complex,G: complex > complex,B4: set_complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ G @ B4 ) )
      = ( groups7754918857620584856omplex
        @ ^ [I3: complex] :
            ( groups7754918857620584856omplex
            @ ^ [J3: complex] : ( times_times_complex @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B4 )
        @ A4 ) ) ).

% sum_product
thf(fact_4327_sum__product,axiom,
    ! [F: nat > nat,A4: set_nat,G: nat > nat,B4: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ B4 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( times_times_nat @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B4 )
        @ A4 ) ) ).

% sum_product
thf(fact_4328_sum__product,axiom,
    ! [F: nat > real,A4: set_nat,G: nat > real,B4: set_nat] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ G @ B4 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( times_times_real @ ( F @ I3 ) @ ( G @ J3 ) )
            @ B4 )
        @ A4 ) ) ).

% sum_product
thf(fact_4329_sum__distrib__right,axiom,
    ! [F: int > int,A4: set_int,R2: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ R2 )
      = ( groups4538972089207619220nt_int
        @ ^ [N3: int] : ( times_times_int @ ( F @ N3 ) @ R2 )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_4330_sum__distrib__right,axiom,
    ! [F: complex > complex,A4: set_complex,R2: complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ R2 )
      = ( groups7754918857620584856omplex
        @ ^ [N3: complex] : ( times_times_complex @ ( F @ N3 ) @ R2 )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_4331_sum__distrib__right,axiom,
    ! [F: nat > nat,A4: set_nat,R2: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ R2 )
      = ( groups3542108847815614940at_nat
        @ ^ [N3: nat] : ( times_times_nat @ ( F @ N3 ) @ R2 )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_4332_sum__distrib__right,axiom,
    ! [F: nat > real,A4: set_nat,R2: real] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ R2 )
      = ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ R2 )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_4333_sum__distrib__left,axiom,
    ! [R2: int,F: int > int,A4: set_int] :
      ( ( times_times_int @ R2 @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N3: int] : ( times_times_int @ R2 @ ( F @ N3 ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_4334_sum__distrib__left,axiom,
    ! [R2: complex,F: complex > complex,A4: set_complex] :
      ( ( times_times_complex @ R2 @ ( groups7754918857620584856omplex @ F @ A4 ) )
      = ( groups7754918857620584856omplex
        @ ^ [N3: complex] : ( times_times_complex @ R2 @ ( F @ N3 ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_4335_sum__distrib__left,axiom,
    ! [R2: nat,F: nat > nat,A4: set_nat] :
      ( ( times_times_nat @ R2 @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N3: nat] : ( times_times_nat @ R2 @ ( F @ N3 ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_4336_sum__distrib__left,axiom,
    ! [R2: real,F: nat > real,A4: set_nat] :
      ( ( times_times_real @ R2 @ ( groups6591440286371151544t_real @ F @ A4 ) )
      = ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( times_times_real @ R2 @ ( F @ N3 ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_4337_sum_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( plus_plus_int @ ( G @ X4 ) @ ( H2 @ X4 ) )
        @ A4 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A4 ) @ ( groups4538972089207619220nt_int @ H2 @ A4 ) ) ) ).

% sum.distrib
thf(fact_4338_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A4: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X4: complex] : ( plus_plus_complex @ ( G @ X4 ) @ ( H2 @ X4 ) )
        @ A4 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A4 ) @ ( groups7754918857620584856omplex @ H2 @ A4 ) ) ) ).

% sum.distrib
thf(fact_4339_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : ( plus_plus_nat @ ( G @ X4 ) @ ( H2 @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A4 ) @ ( groups3542108847815614940at_nat @ H2 @ A4 ) ) ) ).

% sum.distrib
thf(fact_4340_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A4: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X4: nat] : ( plus_plus_real @ ( G @ X4 ) @ ( H2 @ X4 ) )
        @ A4 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A4 ) @ ( groups6591440286371151544t_real @ H2 @ A4 ) ) ) ).

% sum.distrib
thf(fact_4341_sum__subtractf,axiom,
    ! [F: int > int,G: int > int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( minus_minus_int @ ( F @ X4 ) @ ( G @ X4 ) )
        @ A4 )
      = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ).

% sum_subtractf
thf(fact_4342_sum__subtractf,axiom,
    ! [F: complex > complex,G: complex > complex,A4: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X4: complex] : ( minus_minus_complex @ ( F @ X4 ) @ ( G @ X4 ) )
        @ A4 )
      = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ G @ A4 ) ) ) ).

% sum_subtractf
thf(fact_4343_sum__subtractf,axiom,
    ! [F: nat > real,G: nat > real,A4: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X4: nat] : ( minus_minus_real @ ( F @ X4 ) @ ( G @ X4 ) )
        @ A4 )
      = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ G @ A4 ) ) ) ).

% sum_subtractf
thf(fact_4344_sum__divide__distrib,axiom,
    ! [F: complex > complex,A4: set_complex,R2: complex] :
      ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A4 ) @ R2 )
      = ( groups7754918857620584856omplex
        @ ^ [N3: complex] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ R2 )
        @ A4 ) ) ).

% sum_divide_distrib
thf(fact_4345_sum__divide__distrib,axiom,
    ! [F: nat > real,A4: set_nat,R2: real] :
      ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ R2 )
      = ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ R2 )
        @ A4 ) ) ).

% sum_divide_distrib
thf(fact_4346_mod__sum__eq,axiom,
    ! [F: int > int,A: int,A4: set_int] :
      ( ( modulo_modulo_int
        @ ( groups4538972089207619220nt_int
          @ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ A ) ) ).

% mod_sum_eq
thf(fact_4347_mod__sum__eq,axiom,
    ! [F: nat > nat,A: nat,A4: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ A ) ) ).

% mod_sum_eq
thf(fact_4348_sum__nonneg,axiom,
    ! [A4: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4349_sum__nonneg,axiom,
    ! [A4: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4350_sum__nonneg,axiom,
    ! [A4: set_complex,F: complex > real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4351_sum__nonneg,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4352_sum__nonneg,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4353_sum__nonneg,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4354_sum__nonneg,axiom,
    ! [A4: set_complex,F: complex > rat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4355_sum__nonneg,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4356_sum__nonneg,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4357_sum__nonneg,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_4358_sum__nonpos,axiom,
    ! [A4: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_4359_sum__nonpos,axiom,
    ! [A4: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_4360_sum__nonpos,axiom,
    ! [A4: set_complex,F: complex > real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_4361_sum__nonpos,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_4362_sum__nonpos,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_4363_sum__nonpos,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_4364_sum__nonpos,axiom,
    ! [A4: set_complex,F: complex > rat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_4365_sum__nonpos,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_4366_sum__nonpos,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_4367_sum__nonpos,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_4368_sum__mono__inv,axiom,
    ! [F: int > rat,I6: set_int,G: int > rat,I: int] :
      ( ( ( groups3906332499630173760nt_rat @ F @ I6 )
        = ( groups3906332499630173760nt_rat @ G @ I6 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I6 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_int @ I @ I6 )
         => ( ( finite_finite_int @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4369_sum__mono__inv,axiom,
    ! [F: real > rat,I6: set_real,G: real > rat,I: real] :
      ( ( ( groups1300246762558778688al_rat @ F @ I6 )
        = ( groups1300246762558778688al_rat @ G @ I6 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I6 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I6 )
         => ( ( finite_finite_real @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4370_sum__mono__inv,axiom,
    ! [F: nat > rat,I6: set_nat,G: nat > rat,I: nat] :
      ( ( ( groups2906978787729119204at_rat @ F @ I6 )
        = ( groups2906978787729119204at_rat @ G @ I6 ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I6 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_nat @ I @ I6 )
         => ( ( finite_finite_nat @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4371_sum__mono__inv,axiom,
    ! [F: complex > rat,I6: set_complex,G: complex > rat,I: complex] :
      ( ( ( groups5058264527183730370ex_rat @ F @ I6 )
        = ( groups5058264527183730370ex_rat @ G @ I6 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I6 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I6 )
         => ( ( finite3207457112153483333omplex @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4372_sum__mono__inv,axiom,
    ! [F: int > nat,I6: set_int,G: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I6 )
        = ( groups4541462559716669496nt_nat @ G @ I6 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_int @ I @ I6 )
         => ( ( finite_finite_int @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4373_sum__mono__inv,axiom,
    ! [F: real > nat,I6: set_real,G: real > nat,I: real] :
      ( ( ( groups1935376822645274424al_nat @ F @ I6 )
        = ( groups1935376822645274424al_nat @ G @ I6 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I6 )
         => ( ( finite_finite_real @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4374_sum__mono__inv,axiom,
    ! [F: complex > nat,I6: set_complex,G: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I6 )
        = ( groups5693394587270226106ex_nat @ G @ I6 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I6 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I6 )
         => ( ( finite3207457112153483333omplex @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4375_sum__mono__inv,axiom,
    ! [F: real > int,I6: set_real,G: real > int,I: real] :
      ( ( ( groups1932886352136224148al_int @ F @ I6 )
        = ( groups1932886352136224148al_int @ G @ I6 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I6 )
           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I6 )
         => ( ( finite_finite_real @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4376_sum__mono__inv,axiom,
    ! [F: nat > int,I6: set_nat,G: nat > int,I: nat] :
      ( ( ( groups3539618377306564664at_int @ F @ I6 )
        = ( groups3539618377306564664at_int @ G @ I6 ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I6 )
           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_nat @ I @ I6 )
         => ( ( finite_finite_nat @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4377_sum__mono__inv,axiom,
    ! [F: complex > int,I6: set_complex,G: complex > int,I: complex] :
      ( ( ( groups5690904116761175830ex_int @ F @ I6 )
        = ( groups5690904116761175830ex_int @ G @ I6 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I6 )
           => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I6 )
         => ( ( finite3207457112153483333omplex @ I6 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_4378_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_4379_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_4380_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_4381_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_4382_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_4383_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).

% of_bool_less_eq_one
thf(fact_4384_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).

% of_bool_less_eq_one
thf(fact_4385_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).

% of_bool_less_eq_one
thf(fact_4386_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).

% of_bool_less_eq_one
thf(fact_4387_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_4388_of__bool__def,axiom,
    ( zero_n1201886186963655149omplex
    = ( ^ [P5: $o] : ( if_complex @ P5 @ one_one_complex @ zero_zero_complex ) ) ) ).

% of_bool_def
thf(fact_4389_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P5: $o] : ( if_real @ P5 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_4390_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P5: $o] : ( if_rat @ P5 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_4391_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P5: $o] : ( if_nat @ P5 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_4392_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P5: $o] : ( if_int @ P5 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_4393_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P5: $o] : ( if_Code_integer @ P5 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_4394_split__of__bool,axiom,
    ! [P: complex > $o,P2: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
      = ( ( P2
         => ( P @ one_one_complex ) )
        & ( ~ P2
         => ( P @ zero_zero_complex ) ) ) ) ).

% split_of_bool
thf(fact_4395_split__of__bool,axiom,
    ! [P: real > $o,P2: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
      = ( ( P2
         => ( P @ one_one_real ) )
        & ( ~ P2
         => ( P @ zero_zero_real ) ) ) ) ).

% split_of_bool
thf(fact_4396_split__of__bool,axiom,
    ! [P: rat > $o,P2: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
      = ( ( P2
         => ( P @ one_one_rat ) )
        & ( ~ P2
         => ( P @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_4397_split__of__bool,axiom,
    ! [P: nat > $o,P2: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( ( P2
         => ( P @ one_one_nat ) )
        & ( ~ P2
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_4398_split__of__bool,axiom,
    ! [P: int > $o,P2: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = ( ( P2
         => ( P @ one_one_int ) )
        & ( ~ P2
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_4399_split__of__bool,axiom,
    ! [P: code_integer > $o,P2: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
      = ( ( P2
         => ( P @ one_one_Code_integer ) )
        & ( ~ P2
         => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).

% split_of_bool
thf(fact_4400_split__of__bool__asm,axiom,
    ! [P: complex > $o,P2: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_complex ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_complex ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4401_split__of__bool__asm,axiom,
    ! [P: real > $o,P2: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_real ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_real ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4402_split__of__bool__asm,axiom,
    ! [P: rat > $o,P2: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_rat ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4403_split__of__bool__asm,axiom,
    ! [P: nat > $o,P2: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_nat ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4404_split__of__bool__asm,axiom,
    ! [P: int > $o,P2: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_int ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4405_split__of__bool__asm,axiom,
    ! [P: code_integer > $o,P2: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_Code_integer ) )
            | ( ~ P2
              & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4406_finite__set__decode,axiom,
    ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).

% finite_set_decode
thf(fact_4407_replicate__eqI,axiom,
    ! [Xs2: list_real,N: nat,X: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = N )
     => ( ! [Y4: real] :
            ( ( member_real @ Y4 @ ( set_real2 @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replicate_real @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_4408_replicate__eqI,axiom,
    ! [Xs2: list_complex,N: nat,X: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs2 )
        = N )
     => ( ! [Y4: complex] :
            ( ( member_complex @ Y4 @ ( set_complex2 @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replicate_complex @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_4409_replicate__eqI,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs2 )
        = N )
     => ( ! [Y4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ Y4 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replic4235873036481779905at_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_4410_replicate__eqI,axiom,
    ! [Xs2: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = N )
     => ( ! [Y4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replicate_VEBT_VEBT @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_4411_replicate__eqI,axiom,
    ! [Xs2: list_o,N: nat,X: $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = N )
     => ( ! [Y4: $o] :
            ( ( member_o @ Y4 @ ( set_o2 @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replicate_o @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_4412_replicate__eqI,axiom,
    ! [Xs2: list_nat,N: nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = N )
     => ( ! [Y4: nat] :
            ( ( member_nat @ Y4 @ ( set_nat2 @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replicate_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_4413_replicate__eqI,axiom,
    ! [Xs2: list_int,N: nat,X: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = N )
     => ( ! [Y4: int] :
            ( ( member_int @ Y4 @ ( set_int2 @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replicate_int @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_4414_replicate__length__same,axiom,
    ! [Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( X5 = X ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_4415_replicate__length__same,axiom,
    ! [Xs2: list_o,X: $o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
         => ( X5 = X ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_4416_replicate__length__same,axiom,
    ! [Xs2: list_nat,X: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
         => ( X5 = X ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_4417_replicate__length__same,axiom,
    ! [Xs2: list_int,X: int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
         => ( X5 = X ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_4418_sum__le__included,axiom,
    ! [S2: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4419_sum__le__included,axiom,
    ! [S2: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S2 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4420_sum__le__included,axiom,
    ! [S2: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4421_sum__le__included,axiom,
    ! [S2: set_complex,T: set_nat,G: nat > rat,I: nat > complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S2 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4422_sum__le__included,axiom,
    ! [S2: set_complex,T: set_complex,G: complex > rat,I: complex > complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4423_sum__le__included,axiom,
    ! [S2: set_complex,T: set_complex,G: complex > nat,I: complex > complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4424_sum__le__included,axiom,
    ! [S2: set_nat,T: set_nat,G: nat > int,I: nat > nat,F: nat > int] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_less_eq_int @ zero_zero_int @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S2 ) @ ( groups3539618377306564664at_int @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4425_sum__le__included,axiom,
    ! [S2: set_nat,T: set_complex,G: complex > int,I: complex > nat,F: nat > int] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_int @ zero_zero_int @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S2 ) @ ( groups5690904116761175830ex_int @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4426_sum__le__included,axiom,
    ! [S2: set_complex,T: set_nat,G: nat > int,I: nat > complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_less_eq_int @ zero_zero_int @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ S2 ) @ ( groups3539618377306564664at_int @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4427_sum__le__included,axiom,
    ! [S2: set_complex,T: set_complex,G: complex > int,I: complex > complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_int @ zero_zero_int @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ S2 ) @ ( groups5690904116761175830ex_int @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_4428_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4429_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4430_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4431_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4432_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4433_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4434_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4435_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
            = zero_zero_nat )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4436_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ A4 )
            = zero_zero_nat )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4437_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
            = zero_zero_nat )
          = ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_4438_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A4 )
              & ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4439_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
              & ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4440_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A4 )
              & ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4441_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A4 )
              & ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4442_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
              & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4443_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A4 )
              & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4444_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > int,G: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A4 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: int] :
              ( ( member_int @ X3 @ A4 )
              & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4445_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
              & ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4446_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: nat > real,G: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
              & ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_4447_sum_Orelated,axiom,
    ! [R: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X22 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups2073611262835488442omplex @ H2 @ S3 ) @ ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4448_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,X22: real,Y23: real] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X22 @ 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_4449_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups2906978787729119204at_rat @ H2 @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4450_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4451_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,X22: nat,Y23: nat] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X22 @ 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_4452_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,X22: int,Y23: int] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X22 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups3539618377306564664at_int @ H2 @ S3 ) @ ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4453_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,X22: int,Y23: int] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X22 @ 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_4454_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,X22: int,Y23: int] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X22 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups4538972089207619220nt_int @ H2 @ S3 ) @ ( groups4538972089207619220nt_int @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4455_sum_Orelated,axiom,
    ! [R: complex > complex > $o,S3: set_complex,H2: complex > complex,G: complex > complex] :
      ( ( R @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X22 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups7754918857620584856omplex @ H2 @ S3 ) @ ( groups7754918857620584856omplex @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4456_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_nat,H2: nat > nat,G: nat > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X22 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups3542108847815614940at_nat @ H2 @ S3 ) @ ( groups3542108847815614940at_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_4457_sum__strict__mono,axiom,
    ! [A4: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4458_sum__strict__mono,axiom,
    ! [A4: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ A4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4459_sum__strict__mono,axiom,
    ! [A4: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4460_sum__strict__mono,axiom,
    ! [A4: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4461_sum__strict__mono,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4462_sum__strict__mono,axiom,
    ! [A4: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ A4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4463_sum__strict__mono,axiom,
    ! [A4: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4464_sum__strict__mono,axiom,
    ! [A4: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4465_sum__strict__mono,axiom,
    ! [A4: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ A4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4466_sum__strict__mono,axiom,
    ! [A4: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_4467_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ one_one_real @ X ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_4468_ln__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_4469_ln__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_gt_zero
thf(fact_4470_ln__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_ge_zero
thf(fact_4471_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_int,S3: set_int,I: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3049146728041665814omplex @ G @ S3 )
                        = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4472_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_real,S3: set_int,I: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3049146728041665814omplex @ G @ S3 )
                        = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4473_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_int,S3: set_real,I: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5754745047067104278omplex @ G @ S3 )
                        = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4474_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_real,S3: set_real,I: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5754745047067104278omplex @ G @ S3 )
                        = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4475_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_int,S3: set_int,I: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S3 )
                        = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4476_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_real,S3: set_int,I: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S3 )
                        = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4477_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_int,S3: set_real,I: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = zero_zero_real ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S3 )
                        = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4478_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_real,S3: set_real,I: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S3 )
                        = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4479_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_complex,S3: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: complex] :
                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( ( member_complex @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = zero_zero_real ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S3 )
                        = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4480_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_complex,S3: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: complex] :
                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = zero_zero_real ) )
                 => ( ! [B3: complex] :
                        ( ( member_complex @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S3 )
                        = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_4481_sum__nonneg__0,axiom,
    ! [S2: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S2 )
            = zero_zero_real )
         => ( ( member_int @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_4482_sum__nonneg__0,axiom,
    ! [S2: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S2 )
            = zero_zero_real )
         => ( ( member_real @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_4483_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S2 )
            = zero_zero_real )
         => ( ( member_complex @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_4484_sum__nonneg__0,axiom,
    ! [S2: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
            = zero_zero_rat )
         => ( ( member_int @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_4485_sum__nonneg__0,axiom,
    ! [S2: set_real,F: real > rat,I: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S2 )
            = zero_zero_rat )
         => ( ( member_real @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_4486_sum__nonneg__0,axiom,
    ! [S2: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S2 )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_4487_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S2 )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_4488_sum__nonneg__0,axiom,
    ! [S2: set_int,F: int > nat,I: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S2 )
            = zero_zero_nat )
         => ( ( member_int @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_4489_sum__nonneg__0,axiom,
    ! [S2: set_real,F: real > nat,I: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S2 )
            = zero_zero_nat )
         => ( ( member_real @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_4490_sum__nonneg__0,axiom,
    ! [S2: set_complex,F: complex > nat,I: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S2 )
            = zero_zero_nat )
         => ( ( member_complex @ I @ S2 )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_4491_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: int > real,B4: real,I: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S2 )
            = B4 )
         => ( ( member_int @ I @ S2 )
           => ( ord_less_eq_real @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_4492_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: real > real,B4: real,I: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S2 )
            = B4 )
         => ( ( member_real @ I @ S2 )
           => ( ord_less_eq_real @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_4493_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: complex > real,B4: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S2 )
            = B4 )
         => ( ( member_complex @ I @ S2 )
           => ( ord_less_eq_real @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_4494_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: int > rat,B4: rat,I: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
            = B4 )
         => ( ( member_int @ I @ S2 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_4495_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: real > rat,B4: rat,I: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S2 )
            = B4 )
         => ( ( member_real @ I @ S2 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_4496_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F: nat > rat,B4: rat,I: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S2 )
            = B4 )
         => ( ( member_nat @ I @ S2 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_4497_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: complex > rat,B4: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S2 )
            = B4 )
         => ( ( member_complex @ I @ S2 )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_4498_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F: int > nat,B4: nat,I: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S2 )
            = B4 )
         => ( ( member_int @ I @ S2 )
           => ( ord_less_eq_nat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_4499_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F: real > nat,B4: nat,I: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S2 )
            = B4 )
         => ( ( member_real @ I @ S2 )
           => ( ord_less_eq_nat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_4500_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F: complex > nat,B4: nat,I: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S2 )
            = B4 )
         => ( ( member_complex @ I @ S2 )
           => ( ord_less_eq_nat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_4501_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: real > complex] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups5754745047067104278omplex @ G
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = zero_zero_complex ) ) ) )
        = ( groups5754745047067104278omplex @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_4502_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = zero_zero_real ) ) ) )
        = ( groups8097168146408367636l_real @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_4503_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_4504_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1300246762558778688al_rat @ G
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_4505_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_4506_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1935376822645274424al_nat @ G
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = zero_zero_nat ) ) ) )
        = ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_4507_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5693394587270226106ex_nat @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = zero_zero_nat ) ) ) )
        = ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_4508_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: real > int] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1932886352136224148al_int @ G
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = zero_zero_int ) ) ) )
        = ( groups1932886352136224148al_int @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_4509_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5690904116761175830ex_int @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = zero_zero_int ) ) ) )
        = ( groups5690904116761175830ex_int @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_4510_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2073611262835488442omplex @ G
          @ ( minus_minus_set_nat @ A4
            @ ( collect_nat
              @ ^ [X4: nat] :
                  ( ( G @ X4 )
                  = zero_zero_complex ) ) ) )
        = ( groups2073611262835488442omplex @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_4511_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4512_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4513_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 ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4514_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I6 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4515_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I6 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4516_sum__pos2,axiom,
    ! [I6: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( member_nat @ I @ I6 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I6 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4517_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I6 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4518_sum__pos2,axiom,
    ! [I6: set_int,I: int,F: int > nat] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I6 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4519_sum__pos2,axiom,
    ! [I6: set_real,I: real,F: real > nat] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I6 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4520_sum__pos2,axiom,
    ! [I6: set_complex,I: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I6 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_4521_sum__pos,axiom,
    ! [I6: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_4522_sum__pos,axiom,
    ! [I6: set_int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I6 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_4523_sum__pos,axiom,
    ! [I6: set_real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_4524_sum__pos,axiom,
    ! [I6: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_4525_sum__pos,axiom,
    ! [I6: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I6 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_4526_sum__pos,axiom,
    ! [I6: set_int,F: int > rat] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I6 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_4527_sum__pos,axiom,
    ! [I6: set_real,F: real > rat] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_4528_sum__pos,axiom,
    ! [I6: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_4529_sum__pos,axiom,
    ! [I6: set_int,F: int > nat] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_4530_sum__pos,axiom,
    ! [I6: set_real,F: real > nat] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_4531_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_4532_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups3049146728041665814omplex @ G @ A4 )
                  = ( groups3049146728041665814omplex @ H2 @ B4 ) )
                = ( ( groups3049146728041665814omplex @ G @ C4 )
                  = ( groups3049146728041665814omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4533_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ A4 )
                  = ( groups5754745047067104278omplex @ H2 @ B4 ) )
                = ( ( groups5754745047067104278omplex @ G @ C4 )
                  = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4534_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ A4 )
                  = ( groups8778361861064173332t_real @ H2 @ B4 ) )
                = ( ( groups8778361861064173332t_real @ G @ C4 )
                  = ( groups8778361861064173332t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4535_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A4 )
                  = ( groups8097168146408367636l_real @ H2 @ B4 ) )
                = ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4536_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A4 )
                  = ( groups5808333547571424918x_real @ H2 @ B4 ) )
                = ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4537_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ A4 )
                  = ( groups3906332499630173760nt_rat @ H2 @ B4 ) )
                = ( ( groups3906332499630173760nt_rat @ G @ C4 )
                  = ( groups3906332499630173760nt_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4538_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ A4 )
                  = ( groups1300246762558778688al_rat @ H2 @ B4 ) )
                = ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4539_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ A4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B4 ) )
                = ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4540_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups4541462559716669496nt_nat @ G @ A4 )
                  = ( groups4541462559716669496nt_nat @ H2 @ B4 ) )
                = ( ( groups4541462559716669496nt_nat @ G @ C4 )
                  = ( groups4541462559716669496nt_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4541_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ A4 )
                  = ( groups1935376822645274424al_nat @ H2 @ B4 ) )
                = ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_4542_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups3049146728041665814omplex @ G @ C4 )
                  = ( groups3049146728041665814omplex @ H2 @ C4 ) )
               => ( ( groups3049146728041665814omplex @ G @ A4 )
                  = ( groups3049146728041665814omplex @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4543_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ C4 )
                  = ( groups5754745047067104278omplex @ H2 @ C4 ) )
               => ( ( groups5754745047067104278omplex @ G @ A4 )
                  = ( groups5754745047067104278omplex @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4544_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ C4 )
                  = ( groups8778361861064173332t_real @ H2 @ C4 ) )
               => ( ( groups8778361861064173332t_real @ G @ A4 )
                  = ( groups8778361861064173332t_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4545_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) )
               => ( ( groups8097168146408367636l_real @ G @ A4 )
                  = ( groups8097168146408367636l_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4546_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) )
               => ( ( groups5808333547571424918x_real @ G @ A4 )
                  = ( groups5808333547571424918x_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4547_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ C4 )
                  = ( groups3906332499630173760nt_rat @ H2 @ C4 ) )
               => ( ( groups3906332499630173760nt_rat @ G @ A4 )
                  = ( groups3906332499630173760nt_rat @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4548_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
               => ( ( groups1300246762558778688al_rat @ G @ A4 )
                  = ( groups1300246762558778688al_rat @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4549_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
               => ( ( groups5058264527183730370ex_rat @ G @ A4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4550_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups4541462559716669496nt_nat @ G @ C4 )
                  = ( groups4541462559716669496nt_nat @ H2 @ C4 ) )
               => ( ( groups4541462559716669496nt_nat @ G @ A4 )
                  = ( groups4541462559716669496nt_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4551_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
               => ( ( groups1935376822645274424al_nat @ G @ A4 )
                  = ( groups1935376822645274424al_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_4552_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_4553_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ S3 )
            = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4554_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_4555_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_4556_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G @ S3 )
            = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4557_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ S3 )
            = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4558_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ S3 )
            = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4559_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > int] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups4538972089207619220nt_int @ G @ S3 )
            = ( groups4538972089207619220nt_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4560_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups7754918857620584856omplex @ G @ S3 )
            = ( groups7754918857620584856omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4561_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups3542108847815614940at_nat @ G @ S3 )
            = ( groups3542108847815614940at_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_4562_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_4563_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ T3 )
            = ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4564_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_4565_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_4566_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G @ T3 )
            = ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4567_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ T3 )
            = ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4568_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ T3 )
            = ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4569_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > int] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups4538972089207619220nt_int @ G @ T3 )
            = ( groups4538972089207619220nt_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4570_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups7754918857620584856omplex @ G @ T3 )
            = ( groups7754918857620584856omplex @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4571_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups3542108847815614940at_nat @ G @ T3 )
            = ( groups3542108847815614940at_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_4572_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H2: int > complex,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3049146728041665814omplex @ G @ S3 )
              = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4573_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ S3 )
              = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4574_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_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ S3 )
              = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4575_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_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S3 )
              = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4576_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_4577_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ S3 )
              = ( groups3906332499630173760nt_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4578_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ S3 )
              = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4579_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ S3 )
              = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4580_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_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4541462559716669496nt_nat @ G @ S3 )
              = ( groups4541462559716669496nt_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4581_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_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ S3 )
              = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_4582_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3049146728041665814omplex @ G @ T3 )
              = ( groups3049146728041665814omplex @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4583_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ T3 )
              = ( groups5754745047067104278omplex @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4584_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_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ T3 )
              = ( groups8778361861064173332t_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4585_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_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T3 )
              = ( groups8097168146408367636l_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4586_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_4587_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ T3 )
              = ( groups3906332499630173760nt_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4588_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ T3 )
              = ( groups1300246762558778688al_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4589_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ T3 )
              = ( groups5058264527183730370ex_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4590_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_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4541462559716669496nt_nat @ G @ T3 )
              = ( groups4541462559716669496nt_nat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4591_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_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ T3 )
              = ( groups1935376822645274424al_nat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_4592_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5808333547571424918x_real @ G @ A4 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5808333547571424918x_real @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4593_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5058264527183730370ex_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5058264527183730370ex_rat @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4594_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5693394587270226106ex_nat @ G @ A4 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5693394587270226106ex_nat @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4595_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5690904116761175830ex_int @ G @ A4 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5690904116761175830ex_int @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4596_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups2906978787729119204at_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups2906978787729119204at_rat @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4597_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups3539618377306564664at_int @ G @ A4 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups3539618377306564664at_int @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4598_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B4 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups4538972089207619220nt_int @ G @ A4 )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A4 @ B4 ) ) @ ( groups4538972089207619220nt_int @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4599_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > complex] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups7754918857620584856omplex @ G @ A4 )
          = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups7754918857620584856omplex @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4600_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups3542108847815614940at_nat @ G @ A4 )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups3542108847815614940at_nat @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4601_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups6591440286371151544t_real @ G @ A4 )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups6591440286371151544t_real @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_4602_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4603_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4604_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4605_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A4 @ B4 ) )
          = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4606_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ B4 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4607_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_set_int @ B4 @ A4 )
       => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A4 @ B4 ) )
          = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4608_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4609_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A4 @ B4 ) )
          = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_4610_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self
thf(fact_4611_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
          = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_mult
thf(fact_4612_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ln_ln_real @ X )
          = ( minus_minus_real @ X @ one_one_real ) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_4613_ln__div,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
          = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_div
thf(fact_4614_subset__decode__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M2 ) @ ( nat_set_decode @ N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% subset_decode_imp_le
thf(fact_4615_sum__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: int > real] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4616_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > real] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4617_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4618_sum__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: int > rat] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4619_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > rat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4620_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4621_sum__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ A4 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4622_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > nat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4623_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4624_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > int] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
         => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_4625_ln__le__minus__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_4626_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).

% ln_diff_le
thf(fact_4627_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] :
      ( ( zero_n2687167440665602831ol_nat
        @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_4628_of__bool__odd__eq__mod__2,axiom,
    ! [A: int] :
      ( ( zero_n2684676970156552555ol_int
        @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_4629_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] :
      ( ( zero_n356916108424825756nteger
        @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_4630_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K2: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% signed_take_bit_int_less_exp
thf(fact_4631_even__signed__take__bit__iff,axiom,
    ! [M2: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M2 @ A ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_signed_take_bit_iff
thf(fact_4632_even__signed__take__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M2 @ A ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_signed_take_bit_iff
thf(fact_4633_sum__strict__mono2,axiom,
    ! [B4: set_int,A4: set_int,B: int,F: int > real] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B4 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ B4 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4634_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: real > real] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B4 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B4 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4635_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B4 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4636_sum__strict__mono2,axiom,
    ! [B4: set_int,A4: set_int,B: int,F: int > rat] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B4 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ B4 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4637_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: real > rat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B4 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B4 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4638_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B4 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4639_sum__strict__mono2,axiom,
    ! [B4: set_int,A4: set_int,B: int,F: int > nat] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B4 @ A4 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ B4 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4640_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: real > nat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B4 @ A4 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B4 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4641_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B4 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4642_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: real > int] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B4 @ A4 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B4 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
             => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_4643_bits__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [A3: nat] :
          ( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A3 )
         => ( P @ A3 ) )
     => ( ! [A3: nat,B3: $o] :
            ( ( P @ A3 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A3 )
             => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_4644_bits__induct,axiom,
    ! [P: int > $o,A: int] :
      ( ! [A3: int] :
          ( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A3 )
         => ( P @ A3 ) )
     => ( ! [A3: int,B3: $o] :
            ( ( P @ A3 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A3 )
             => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_4645_bits__induct,axiom,
    ! [P: code_integer > $o,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A3 )
         => ( P @ A3 ) )
     => ( ! [A3: code_integer,B3: $o] :
            ( ( P @ A3 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A3 )
             => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_4646_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_eq_int @ K2 @ ( bit_ri631733984087533419it_int @ N @ K2 ) )
      = ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_4647_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ K2 )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_4648_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B
        = ( plus_plus_complex @ B @ A ) )
      = ( A = zero_zero_complex ) ) ).

% add_0_iff
thf(fact_4649_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_4650_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_4651_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_4652_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_4653_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M2 @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_4654_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M2 @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_4655_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M2 @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_4656_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_4657_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_4658_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_4659_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_4660_crossproduct__eq,axiom,
    ! [W: real,Y: real,X: real,Z: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z ) )
        = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_4661_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
        = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_4662_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_4663_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_4664_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ ( minus_minus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_4665_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_4666_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X4: nat] :
          ( collect_nat
          @ ^ [N3: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_4667_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4668_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4669_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger
        @ ( zero_n356916108424825756nteger
          @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 )
             != zero_z3403309356797280102nteger )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4670_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_4671_ln__2__less__1,axiom,
    ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).

% ln_2_less_1
thf(fact_4672_tanh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( tanh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_4673_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_4674_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N3: nat,A5: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4675_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,A5: int] : ( if_int @ ( N3 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4676_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_4677_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 ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_4678_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_4679_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_4680_divmod__step__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_4681_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_4682_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_4683_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = ( uminus1482373934393186551omplex @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4684_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4685_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4686_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4687_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4688_add_Oinverse__inverse,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4689_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4690_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4691_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_4692_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_4693_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_4694_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_4695_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_4696_abs__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_abs
thf(fact_4697_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_4698_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4699_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4700_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_4701_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_4702_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_4703_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_4704_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_4705_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_4706_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_4707_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_4708_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_4709_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_4710_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_4711_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_4712_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_4713_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
thf(fact_4714_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_4715_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_4716_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_4717_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4718_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( uminus_uminus_rat @ A ) )
      = ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4719_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4720_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_4721_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_4722_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_4723_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_4724_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_4725_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_4726_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_4727_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4728_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4729_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_4730_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_4731_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_4732_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_4733_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_4734_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_4735_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_4736_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_4737_mult__minus__left,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4738_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_4739_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_4740_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_4741_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_4742_minus__mult__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( times_times_rat @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4743_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( times_3573771949741848930nteger @ A @ B ) ) ).

% minus_mult_minus
thf(fact_4744_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_4745_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_4746_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_4747_mult__minus__right,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4748_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_4749_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_4750_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_4751_add__minus__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4752_add__minus__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4753_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4754_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_4755_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_4756_minus__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4757_minus__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4758_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4759_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_4760_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_4761_minus__add__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_add_distrib
thf(fact_4762_minus__add__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_add_distrib
thf(fact_4763_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_add_distrib
thf(fact_4764_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_4765_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_4766_minus__diff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
      = ( minus_minus_complex @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4767_minus__diff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
      = ( minus_minus_rat @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4768_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4769_div__minus__minus,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( divide_divide_int @ A @ B ) ) ).

% div_minus_minus
thf(fact_4770_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ A @ B ) ) ).

% div_minus_minus
thf(fact_4771_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_4772_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_4773_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_4774_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_4775_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_4776_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_4777_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_4778_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_4779_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_4780_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_4781_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_4782_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_4783_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_4784_abs__0,axiom,
    ( ( abs_abs_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% abs_0
thf(fact_4785_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_4786_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_4787_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_4788_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_4789_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_4790_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4791_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y )
      = ( dvd_dvd_rat @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4792_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y )
      = ( dvd_dvd_Code_integer @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4793_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_4794_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_4795_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y ) )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4796_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y ) )
      = ( dvd_dvd_rat @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4797_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y ) )
      = ( dvd_dvd_Code_integer @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4798_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_numeral
thf(fact_4799_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_numeral
thf(fact_4800_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_numeral
thf(fact_4801_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_numeral
thf(fact_4802_abs__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_4803_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_4804_abs__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
      = ( times_times_rat @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_4805_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_4806_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_4807_abs__1,axiom,
    ( ( abs_abs_complex @ one_one_complex )
    = one_one_complex ) ).

% abs_1
thf(fact_4808_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_4809_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_4810_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_4811_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_add_abs
thf(fact_4812_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_4813_abs__add__abs,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_add_abs
thf(fact_4814_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_4815_abs__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).

% abs_divide
thf(fact_4816_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_4817_abs__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_divide
thf(fact_4818_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_4819_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_4820_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_4821_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_4822_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_4823_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_4824_abs__minus,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( abs_abs_complex @ A ) ) ).

% abs_minus
thf(fact_4825_abs__minus,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus
thf(fact_4826_abs__minus,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus
thf(fact_4827_mod__minus__minus,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_4828_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_4829_dvd__abs__iff,axiom,
    ! [M2: real,K2: real] :
      ( ( dvd_dvd_real @ M2 @ ( abs_abs_real @ K2 ) )
      = ( dvd_dvd_real @ M2 @ K2 ) ) ).

% dvd_abs_iff
thf(fact_4830_dvd__abs__iff,axiom,
    ! [M2: int,K2: int] :
      ( ( dvd_dvd_int @ M2 @ ( abs_abs_int @ K2 ) )
      = ( dvd_dvd_int @ M2 @ K2 ) ) ).

% dvd_abs_iff
thf(fact_4831_dvd__abs__iff,axiom,
    ! [M2: code_integer,K2: code_integer] :
      ( ( dvd_dvd_Code_integer @ M2 @ ( abs_abs_Code_integer @ K2 ) )
      = ( dvd_dvd_Code_integer @ M2 @ K2 ) ) ).

% dvd_abs_iff
thf(fact_4832_abs__dvd__iff,axiom,
    ! [M2: real,K2: real] :
      ( ( dvd_dvd_real @ ( abs_abs_real @ M2 ) @ K2 )
      = ( dvd_dvd_real @ M2 @ K2 ) ) ).

% abs_dvd_iff
thf(fact_4833_abs__dvd__iff,axiom,
    ! [M2: int,K2: int] :
      ( ( dvd_dvd_int @ ( abs_abs_int @ M2 ) @ K2 )
      = ( dvd_dvd_int @ M2 @ K2 ) ) ).

% abs_dvd_iff
thf(fact_4834_abs__dvd__iff,axiom,
    ! [M2: code_integer,K2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M2 ) @ K2 )
      = ( dvd_dvd_Code_integer @ M2 @ K2 ) ) ).

% abs_dvd_iff
thf(fact_4835_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% abs_bool_eq
thf(fact_4836_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% abs_bool_eq
thf(fact_4837_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% abs_bool_eq
thf(fact_4838_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_4839_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_4840_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_4841_case__prod__conv,axiom,
    ! [F: int > int > product_prod_int_int,A: int,B: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_4842_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_4843_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_4844_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4845_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4846_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_4847_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_4848_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_4849_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% less_eq_neg_nonpos
thf(fact_4850_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_4851_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_4852_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_le_0_iff_le
thf(fact_4853_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_le_0_iff_le
thf(fact_4854_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_4855_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_4856_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_le_iff_le
thf(fact_4857_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% neg_0_le_iff_le
thf(fact_4858_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_4859_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_4860_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_4861_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_0_iff_less
thf(fact_4862_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_0_iff_less
thf(fact_4863_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_4864_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_4865_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% neg_0_less_iff_less
thf(fact_4866_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_less_iff_less
thf(fact_4867_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_4868_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_4869_neg__less__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_pos
thf(fact_4870_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_4871_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_4872_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_4873_less__neg__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% less_neg_neg
thf(fact_4874_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_4875_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_4876_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_4877_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_4878_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_4879_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_4880_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_4881_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_4882_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_4883_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_4884_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_4885_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_4886_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_4887_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_4888_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_4889_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_4890_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_4891_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_4892_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4893_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4894_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4895_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4896_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4897_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4898_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4899_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4900_mult__minus1__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1_right
thf(fact_4901_mult__minus1__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1_right
thf(fact_4902_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_4903_mult__minus1__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1_right
thf(fact_4904_mult__minus1__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1_right
thf(fact_4905_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_4906_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_4907_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_4908_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_4909_mult__minus1,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1
thf(fact_4910_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
thf(fact_4911_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_4912_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
thf(fact_4913_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_4914_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% abs_le_self_iff
thf(fact_4915_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_4916_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% abs_le_self_iff
thf(fact_4917_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_4918_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4919_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4920_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4921_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4922_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
      = ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
thf(fact_4923_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_4924_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
thf(fact_4925_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_4926_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_4927_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_4928_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( plus_plus_complex @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4929_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( plus_plus_rat @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4930_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( plus_p5714425477246183910nteger @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4931_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_4932_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_4933_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( minus_minus_complex @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4934_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( minus_minus_rat @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4935_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4936_div__minus1__right,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A ) ) ).

% div_minus1_right
thf(fact_4937_div__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% div_minus1_right
thf(fact_4938_divide__minus1,axiom,
    ! [X: real] :
      ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X ) ) ).

% divide_minus1
thf(fact_4939_divide__minus1,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% divide_minus1
thf(fact_4940_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_4941_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_neg_numeral
thf(fact_4942_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_neg_numeral
thf(fact_4943_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_neg_numeral
thf(fact_4944_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_neg_numeral
thf(fact_4945_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_4946_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_4947_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_4948_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_4949_minus__mod__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_mod_self1
thf(fact_4950_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% minus_mod_self1
thf(fact_4951_abs__power__minus,axiom,
    ! [A: real,N: nat] :
      ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
      = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_4952_abs__power__minus,axiom,
    ! [A: int,N: nat] :
      ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
      = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_4953_abs__power__minus,axiom,
    ! [A: rat,N: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
      = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_4954_abs__power__minus,axiom,
    ! [A: code_integer,N: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
      = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_4955_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% signed_take_bit_of_minus_1
thf(fact_4956_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_take_bit_of_minus_1
thf(fact_4957_real__add__minus__iff,axiom,
    ! [X: real,A: real] :
      ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X = A ) ) ).

% real_add_minus_iff
thf(fact_4958_sum__abs,axiom,
    ! [F: int > int,A4: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
        @ A4 ) ) ).

% sum_abs
thf(fact_4959_sum__abs,axiom,
    ! [F: nat > real,A4: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
        @ A4 ) ) ).

% sum_abs
thf(fact_4960_dbl__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) )
      = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K2 ) ) ) ) ).

% dbl_simps(1)
thf(fact_4961_dbl__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).

% dbl_simps(1)
thf(fact_4962_dbl__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K2 ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K2 ) ) ) ) ).

% dbl_simps(1)
thf(fact_4963_dbl__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) ) )
      = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K2 ) ) ) ) ).

% dbl_simps(1)
thf(fact_4964_dbl__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K2 ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K2 ) ) ) ) ).

% dbl_simps(1)
thf(fact_4965_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_4966_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_4967_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_4968_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = zero_zero_rat ) ).

% add_neg_numeral_special(8)
thf(fact_4969_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(8)
thf(fact_4970_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_4971_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_4972_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_4973_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% add_neg_numeral_special(7)
thf(fact_4974_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(7)
thf(fact_4975_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_4976_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_4977_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_4978_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% diff_numeral_special(12)
thf(fact_4979_diff__numeral__special_I12_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(12)
thf(fact_4980_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_real @ one_one_real )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4981_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4982_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4983_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_rat @ one_one_rat )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4984_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4985_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4986_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4987_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4988_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4989_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4990_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_4991_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
      = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
thf(fact_4992_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_4993_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_4994_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_4995_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_4996_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_nonpos
thf(fact_4997_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_4998_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
      = one_one_real ) ).

% minus_one_mult_self
thf(fact_4999_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_5000_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
      = one_one_complex ) ).

% minus_one_mult_self
thf(fact_5001_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_5002_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_5003_left__minus__one__mult__self,axiom,
    ! [N: nat,A: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_5004_left__minus__one__mult__self,axiom,
    ! [N: nat,A: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_5005_left__minus__one__mult__self,axiom,
    ! [N: nat,A: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_5006_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_5007_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_5008_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_5009_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_5010_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_5011_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_5012_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_5013_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_5014_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_5015_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_5016_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_5017_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_5018_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_5019_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_5020_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_5021_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_5022_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_5023_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_5024_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_5025_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_5026_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_5027_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5028_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5029_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5030_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5031_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5032_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5033_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5034_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5035_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5036_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5037_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5038_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5039_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5040_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5041_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5042_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5043_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5044_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5045_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5046_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5047_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5048_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5049_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5050_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5051_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5052_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_5053_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_5054_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_5055_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_5056_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_5057_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_5058_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_5059_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_5060_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_5061_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_5062_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_5063_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_5064_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_5065_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_5066_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_5067_artanh__minus__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
        = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).

% artanh_minus_real
thf(fact_5068_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_eq_num @ N @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_5069_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_eq_num @ N @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_5070_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_eq_num @ N @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_5071_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_eq_num @ N @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_5072_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_num @ N @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_5073_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_num @ N @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_5074_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_num @ N @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_5075_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_num @ N @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_5076_sum__abs__ge__zero,axiom,
    ! [F: int > int,A4: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
        @ A4 ) ) ).

% sum_abs_ge_zero
thf(fact_5077_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A4: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
        @ A4 ) ) ).

% sum_abs_ge_zero
thf(fact_5078_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5079_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5080_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5081_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5082_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_5083_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_5084_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_5085_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_5086_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5087_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
        = A )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5088_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5089_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
            = B ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5090_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
            = B ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5091_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
            = B ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5092_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5093_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5094_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5095_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5096_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_5097_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_5098_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_5099_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_5100_power2__minus,axiom,
    ! [A: real] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_5101_power2__minus,axiom,
    ! [A: int] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_5102_power2__minus,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_5103_power2__minus,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_5104_power2__minus,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_5105_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
      = ( ( A != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5106_zero__less__power__abs__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
      = ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5107_zero__less__power__abs__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
      = ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5108_zero__less__power__abs__iff,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
      = ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5109_abs__power2,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_5110_abs__power2,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_5111_abs__power2,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_5112_power2__abs,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_5113_power2__abs,axiom,
    ! [A: real] :
      ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_5114_power2__abs,axiom,
    ! [A: int] :
      ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_5115_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5116_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5117_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5118_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5119_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5120_sum__zero__power,axiom,
    ! [A4: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A4 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
            @ A4 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_5121_sum__zero__power,axiom,
    ! [A4: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
            @ A4 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
            @ A4 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_5122_sum__zero__power,axiom,
    ! [A4: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A4 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
            @ A4 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_5123_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_5124_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_5125_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_5126_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_5127_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_5128_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_5129_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_5130_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_5131_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_5132_diff__numeral__special_I11_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_5133_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_5134_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_5135_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_5136_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_5137_diff__numeral__special_I10_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_5138_minus__1__div__2__eq,axiom,
    ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_2_eq
thf(fact_5139_minus__1__div__2__eq,axiom,
    ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% minus_1_div_2_eq
thf(fact_5140_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_minus_1_mod_2_eq
thf(fact_5141_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_minus_1_mod_2_eq
thf(fact_5142_minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% minus_1_mod_2_eq
thf(fact_5143_minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% minus_1_mod_2_eq
thf(fact_5144_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5145_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5146_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5147_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5148_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5149_power__minus__odd,axiom,
    ! [N: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
        = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5150_power__minus__odd,axiom,
    ! [N: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
        = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5151_power__minus__odd,axiom,
    ! [N: nat,A: complex] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
        = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5152_power__minus__odd,axiom,
    ! [N: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
        = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5153_power__minus__odd,axiom,
    ! [N: nat,A: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5154_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
        = ( power_power_real @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5155_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
        = ( power_power_int @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5156_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: complex] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
        = ( power_power_complex @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5157_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
        = ( power_power_rat @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5158_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
        = ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5159_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_5160_power__even__abs__numeral,axiom,
    ! [W: num,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_5161_power__even__abs__numeral,axiom,
    ! [W: num,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_5162_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_5163_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_5164_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_5165_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_5166_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_5167_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5168_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5169_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ one_one_complex )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5170_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5171_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5172_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_5173_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5174_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5175_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5176_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5177_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5178_sum__zero__power_H,axiom,
    ! [A4: set_nat,C: nat > complex,D: nat > complex] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
            @ A4 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
            @ A4 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_5179_sum__zero__power_H,axiom,
    ! [A4: set_nat,C: nat > rat,D: nat > rat] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
            @ A4 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
            @ A4 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_5180_sum__zero__power_H,axiom,
    ! [A4: set_nat,C: nat > real,D: nat > real] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
            @ A4 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
            @ A4 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_5181_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_real ) ).

% power_minus1_even
thf(fact_5182_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_5183_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_complex ) ).

% power_minus1_even
thf(fact_5184_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_5185_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_5186_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% neg_one_odd_power
thf(fact_5187_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_5188_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% neg_one_odd_power
thf(fact_5189_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_5190_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_5191_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
        = one_one_real ) ) ).

% neg_one_even_power
thf(fact_5192_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_5193_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
        = one_one_complex ) ) ).

% neg_one_even_power
thf(fact_5194_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_5195_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_5196_signed__take__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_5197_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_5198_signed__take__bit__minus,axiom,
    ! [N: nat,K2: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K2 ) ) ) ).

% signed_take_bit_minus
thf(fact_5199_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_5200_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_5201_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_5202_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_5203_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_5204_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_5205_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_5206_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_5207_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_5208_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_5209_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_5210_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_5211_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_5212_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_5213_abs__ge__minus__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).

% abs_ge_minus_self
thf(fact_5214_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_5215_prod_Ocase__distrib,axiom,
    ! [H2: nat > nat,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( produc6842872674320459806at_nat
        @ ^ [X16: nat,X23: nat] : ( H2 @ ( F @ X16 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5216_prod_Ocase__distrib,axiom,
    ! [H2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X16: int,X23: int] : ( H2 @ ( F @ X16 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5217_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X16: int,X23: int] : ( H2 @ ( F @ X16 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5218_prod_Ocase__distrib,axiom,
    ! [H2: $o > product_prod_int_int,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X16: int,X23: int] : ( H2 @ ( F @ X16 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5219_prod_Ocase__distrib,axiom,
    ! [H2: nat > product_prod_nat_nat > $o,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X16: nat,X23: nat] : ( H2 @ ( F @ X16 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5220_prod_Ocase__distrib,axiom,
    ! [H2: ( product_prod_nat_nat > $o ) > nat,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( produc6842872674320459806at_nat
        @ ^ [X16: nat,X23: nat] : ( H2 @ ( F @ X16 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5221_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_int_int > product_prod_int_int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X16: int,X23: int] : ( H2 @ ( F @ X16 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5222_prod_Ocase__distrib,axiom,
    ! [H2: nat > product_prod_nat_nat > product_prod_nat_nat,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( produc27273713700761075at_nat
        @ ^ [X16: nat,X23: nat] : ( H2 @ ( F @ X16 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5223_prod_Ocase__distrib,axiom,
    ! [H2: ( product_prod_nat_nat > product_prod_nat_nat ) > nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc27273713700761075at_nat @ F @ Prod ) )
      = ( produc6842872674320459806at_nat
        @ ^ [X16: nat,X23: nat] : ( H2 @ ( F @ X16 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5224_prod_Ocase__distrib,axiom,
    ! [H2: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X16: nat,X23: nat] : ( H2 @ ( F @ X16 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5225_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5226_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5227_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( uminus1482373934393186551omplex @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5228_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5229_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( uminus1351360451143612070nteger @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5230_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_5231_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_5232_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% equation_minus_iff
thf(fact_5233_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_5234_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_5235_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_5236_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_5237_abs__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( abs_abs_rat @ X )
        = ( abs_abs_rat @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_5238_abs__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( abs_abs_Code_integer @ X )
        = ( abs_abs_Code_integer @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_5239_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_5240_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_5241_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_rat @ A @ B )
        & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_5242_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le6747313008572928689nteger @ A @ B )
        & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_5243_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_5244_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_5245_abs__minus__le__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).

% abs_minus_le_zero
thf(fact_5246_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_5247_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_5248_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = B )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
        & ( ( A = B )
          | ( A
            = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_5249_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ( abs_abs_rat @ A )
        = B )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_rat @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_5250_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_5251_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_5252_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( abs_abs_Code_integer @ B ) )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
        & ( ( B = A )
          | ( B
            = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_5253_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( abs_abs_rat @ B ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_rat @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_5254_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_5255_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_5256_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_5257_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_5258_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_5259_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_5260_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_5261_abs__of__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_neg
thf(fact_5262_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_5263_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_5264_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_5265_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_5266_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_5267_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).

% abs_real_def
thf(fact_5268_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_5269_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% abs_le_D1
thf(fact_5270_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% abs_le_D1
thf(fact_5271_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_5272_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_5273_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_5274_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_5275_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_5276_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_5277_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( abs_abs_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
thf(fact_5278_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_5279_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_5280_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_5281_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_mult
thf(fact_5282_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_5283_abs__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_mult
thf(fact_5284_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_5285_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_5286_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_5287_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_5288_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_5289_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5290_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_5291_abs__minus__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5292_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_5293_tanh__real__gt__neg1,axiom,
    ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).

% tanh_real_gt_neg1
thf(fact_5294_power__abs,axiom,
    ! [A: code_integer,N: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).

% power_abs
thf(fact_5295_power__abs,axiom,
    ! [A: real,N: nat] :
      ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
      = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).

% power_abs
thf(fact_5296_power__abs,axiom,
    ! [A: int,N: nat] :
      ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
      = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).

% power_abs
thf(fact_5297_dvd__if__abs__eq,axiom,
    ! [L: real,K2: real] :
      ( ( ( abs_abs_real @ L )
        = ( abs_abs_real @ K2 ) )
     => ( dvd_dvd_real @ L @ K2 ) ) ).

% dvd_if_abs_eq
thf(fact_5298_dvd__if__abs__eq,axiom,
    ! [L: int,K2: int] :
      ( ( ( abs_abs_int @ L )
        = ( abs_abs_int @ K2 ) )
     => ( dvd_dvd_int @ L @ K2 ) ) ).

% dvd_if_abs_eq
thf(fact_5299_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K2: code_integer] :
      ( ( ( abs_abs_Code_integer @ L )
        = ( abs_abs_Code_integer @ K2 ) )
     => ( dvd_dvd_Code_integer @ L @ K2 ) ) ).

% dvd_if_abs_eq
thf(fact_5300_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_5301_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_minus_iff
thf(fact_5302_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% le_minus_iff
thf(fact_5303_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_5304_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_5305_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5306_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5307_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_5308_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_5309_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5310_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5311_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_5312_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_5313_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_5314_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_5315_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_5316_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_5317_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% less_minus_iff
thf(fact_5318_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% less_minus_iff
thf(fact_5319_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_5320_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_5321_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5322_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5323_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_5324_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_5325_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5326_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5327_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) )
     != ( numeral_numeral_real @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5328_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
     != ( numeral_numeral_int @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5329_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) )
     != ( numera6690914467698888265omplex @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5330_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) )
     != ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5331_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) )
     != ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5332_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numeral_numeral_real @ M2 )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5333_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numeral_numeral_int @ M2 )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5334_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numera6690914467698888265omplex @ M2 )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5335_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numeral_numeral_rat @ M2 )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5336_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numera6620942414471956472nteger @ M2 )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5337_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_5338_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_5339_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_5340_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ A )
        = ( times_times_rat @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5341_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ A )
        = ( times_3573771949741848930nteger @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1351360451143612070nteger @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5342_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_5343_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_5344_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_5345_minus__mult__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_mult_commute
thf(fact_5346_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_mult_commute
thf(fact_5347_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_5348_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_5349_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_5350_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_5351_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_5352_group__cancel_Oneg1,axiom,
    ! [A4: real,K2: real,A: real] :
      ( ( A4
        = ( plus_plus_real @ K2 @ A ) )
     => ( ( uminus_uminus_real @ A4 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K2 ) @ ( uminus_uminus_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5353_group__cancel_Oneg1,axiom,
    ! [A4: int,K2: int,A: int] :
      ( ( A4
        = ( plus_plus_int @ K2 @ A ) )
     => ( ( uminus_uminus_int @ A4 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5354_group__cancel_Oneg1,axiom,
    ! [A4: complex,K2: complex,A: complex] :
      ( ( A4
        = ( plus_plus_complex @ K2 @ A ) )
     => ( ( uminus1482373934393186551omplex @ A4 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5355_group__cancel_Oneg1,axiom,
    ! [A4: rat,K2: rat,A: rat] :
      ( ( A4
        = ( plus_plus_rat @ K2 @ A ) )
     => ( ( uminus_uminus_rat @ A4 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K2 ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5356_group__cancel_Oneg1,axiom,
    ! [A4: code_integer,K2: code_integer,A: code_integer] :
      ( ( A4
        = ( plus_p5714425477246183910nteger @ K2 @ A ) )
     => ( ( uminus1351360451143612070nteger @ A4 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5357_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_5358_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_5359_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5360_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5361_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5362_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_5363_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_5364_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5365_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5366_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5367_minus__diff__minus,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5368_minus__diff__minus,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5369_minus__diff__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5370_minus__diff__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5371_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5372_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_5373_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_5374_minus__diff__commute,axiom,
    ! [B: complex,A: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
      = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5375_minus__diff__commute,axiom,
    ! [B: rat,A: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5376_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
      = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5377_div__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% div_minus_right
thf(fact_5378_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% div_minus_right
thf(fact_5379_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_5380_minus__divide__right,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_divide_right
thf(fact_5381_minus__divide__right,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_divide_right
thf(fact_5382_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_5383_minus__divide__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( divide1717551699836669952omplex @ A @ B ) ) ).

% minus_divide_divide
thf(fact_5384_minus__divide__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( divide_divide_rat @ A @ B ) ) ).

% minus_divide_divide
thf(fact_5385_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_5386_minus__divide__left,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_5387_minus__divide__left,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_5388_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > nat,X1: nat,X2: nat] :
      ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_5389_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X2: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_5390_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X2: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_5391_old_Oprod_Ocase,axiom,
    ! [F: int > int > product_prod_int_int,X1: int,X2: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_5392_old_Oprod_Ocase,axiom,
    ! [F: int > int > $o,X1: int,X2: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_5393_mod__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_5394_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_5395_mod__minus__cong,axiom,
    ! [A: int,B: int,A2: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = ( modulo_modulo_int @ A2 @ B ) )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
        = ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B ) ) ) ).

% mod_minus_cong
thf(fact_5396_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = ( modulo364778990260209775nteger @ A2 @ B ) )
     => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B ) ) ) ).

% mod_minus_cong
thf(fact_5397_mod__minus__eq,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% mod_minus_eq
thf(fact_5398_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% mod_minus_eq
thf(fact_5399_cond__case__prod__eta,axiom,
    ! [F: nat > nat > nat,G: product_prod_nat_nat > nat] :
      ( ! [X5: nat,Y4: nat] :
          ( ( F @ X5 @ Y4 )
          = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
     => ( ( produc6842872674320459806at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_5400_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X5: nat,Y4: nat] :
          ( ( F @ X5 @ Y4 )
          = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
     => ( ( produc27273713700761075at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_5401_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X5: nat,Y4: nat] :
          ( ( F @ X5 @ Y4 )
          = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
     => ( ( produc8739625826339149834_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_5402_cond__case__prod__eta,axiom,
    ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
      ( ! [X5: int,Y4: int] :
          ( ( F @ X5 @ Y4 )
          = ( G @ ( product_Pair_int_int @ X5 @ Y4 ) ) )
     => ( ( produc4245557441103728435nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_5403_cond__case__prod__eta,axiom,
    ! [F: int > int > $o,G: product_prod_int_int > $o] :
      ( ! [X5: int,Y4: int] :
          ( ( F @ X5 @ Y4 )
          = ( G @ ( product_Pair_int_int @ X5 @ Y4 ) ) )
     => ( ( produc4947309494688390418_int_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_5404_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > nat] :
      ( ( produc6842872674320459806at_nat
        @ ^ [X4: nat,Y3: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5405_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y3: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5406_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( produc8739625826339149834_nat_o
        @ ^ [X4: nat,Y3: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5407_case__prod__eta,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int
        @ ^ [X4: int,Y3: int] : ( F @ ( product_Pair_int_int @ X4 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5408_case__prod__eta,axiom,
    ! [F: product_prod_int_int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [X4: int,Y3: int] : ( F @ ( product_Pair_int_int @ X4 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5409_case__prodE2,axiom,
    ! [Q: nat > $o,P: nat > nat > nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6842872674320459806at_nat @ P @ Z ) )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_5410_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc27273713700761075at_nat @ P @ Z ) )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_5411_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z ) )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_5412_case__prodE2,axiom,
    ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
      ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z ) )
     => ~ ! [X5: int,Y4: int] :
            ( ( Z
              = ( product_Pair_int_int @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_5413_case__prodE2,axiom,
    ! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
      ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
     => ~ ! [X5: int,Y4: int] :
            ( ( Z
              = ( product_Pair_int_int @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_5414_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_5415_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_5416_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_5417_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_5418_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_5419_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_5420_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_5421_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_5422_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5423_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5424_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5425_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5426_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_5427_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_5428_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_5429_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_5430_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5431_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_5432_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_5433_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_5434_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5435_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_5436_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_5437_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_5438_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_5439_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_5440_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_5441_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_5442_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_5443_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_5444_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_5445_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_5446_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_5447_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% nonzero_abs_divide
thf(fact_5448_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ ( suc @ X5 ) @ A4 )
           => ( ( F @ ( suc @ X5 ) )
              = ( G @ ( suc @ X5 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A4 )
          = ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).

% sum_cong_Suc
thf(fact_5449_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ ( suc @ X5 ) @ A4 )
           => ( ( F @ ( suc @ X5 ) )
              = ( G @ ( suc @ X5 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A4 )
          = ( groups6591440286371151544t_real @ G @ A4 ) ) ) ) ).

% sum_cong_Suc
thf(fact_5450_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5451_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5452_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5453_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5454_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5455_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5456_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5457_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5458_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5459_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5460_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5461_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5462_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5463_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5464_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5465_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5466_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5467_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5468_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5469_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5470_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5471_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_5472_le__minus__one__simps_I2_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% le_minus_one_simps(2)
thf(fact_5473_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% le_minus_one_simps(2)
thf(fact_5474_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_5475_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_5476_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(4)
thf(fact_5477_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(4)
thf(fact_5478_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_5479_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_5480_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_5481_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_5482_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_5483_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_5484_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_5485_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_5486_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_5487_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5488_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5489_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_5490_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_5491_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_5492_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5493_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5494_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_5495_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_5496_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5497_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5498_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5499_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_5500_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_5501_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_5502_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
thf(fact_5503_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_group_add_class.ab_left_minus
thf(fact_5504_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_5505_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_5506_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_5507_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5508_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5509_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_5510_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_5511_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(4)
thf(fact_5512_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(4)
thf(fact_5513_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_5514_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_5515_less__minus__one__simps_I2_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% less_minus_one_simps(2)
thf(fact_5516_less__minus__one__simps_I2_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% less_minus_one_simps(2)
thf(fact_5517_numeral__times__minus__swap,axiom,
    ! [W: num,X: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
      = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5518_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
      = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5519_numeral__times__minus__swap,axiom,
    ! [W: num,X: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
      = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5520_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
      = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5521_numeral__times__minus__swap,axiom,
    ! [W: num,X: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
      = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5522_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_5523_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_5524_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_5525_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_5526_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_5527_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_5528_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5529_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5530_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5531_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5532_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5533_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ N )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_5534_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ N )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_5535_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ N )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_5536_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ N )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_5537_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ N )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_5538_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_5539_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_5540_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_5541_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( times_times_rat @ X @ X )
        = one_one_rat )
      = ( ( X = one_one_rat )
        | ( X
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_5542_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X @ X )
        = one_one_Code_integer )
      = ( ( X = one_one_Code_integer )
        | ( X
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_5543_group__cancel_Osub2,axiom,
    ! [B4: real,K2: real,B: real,A: real] :
      ( ( B4
        = ( plus_plus_real @ K2 @ B ) )
     => ( ( minus_minus_real @ A @ B4 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K2 ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5544_group__cancel_Osub2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4
        = ( plus_plus_int @ K2 @ B ) )
     => ( ( minus_minus_int @ A @ B4 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5545_group__cancel_Osub2,axiom,
    ! [B4: complex,K2: complex,B: complex,A: complex] :
      ( ( B4
        = ( plus_plus_complex @ K2 @ B ) )
     => ( ( minus_minus_complex @ A @ B4 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K2 ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5546_group__cancel_Osub2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4
        = ( plus_plus_rat @ K2 @ B ) )
     => ( ( minus_minus_rat @ A @ B4 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K2 ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5547_group__cancel_Osub2,axiom,
    ! [B4: code_integer,K2: code_integer,B: code_integer,A: code_integer] :
      ( ( B4
        = ( plus_p5714425477246183910nteger @ K2 @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B4 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K2 ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5548_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A5: real,B5: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B5 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5549_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A5: int,B5: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B5 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5550_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A5: complex,B5: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B5 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5551_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A5: rat,B5: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B5 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5552_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A5: code_integer,B5: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B5 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5553_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A5: real,B5: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B5 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5554_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A5: int,B5: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B5 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5555_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A5: complex,B5: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B5 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5556_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A5: rat,B5: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B5 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5557_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A5: code_integer,B5: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B5 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5558_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_5559_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_5560_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_5561_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_5562_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_5563_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_5564_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_5565_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_5566_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_5567_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_5568_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_5569_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).

% real_minus_mult_self_le
thf(fact_5570_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M2: int,N: int] :
      ( ( ( times_times_int @ M2 @ N )
        = one_one_int )
     => ( ( M2 = one_one_int )
        | ( M2
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_5571_zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( ( ( times_times_int @ M2 @ N )
        = one_one_int )
      = ( ( ( M2 = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M2
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_5572_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_5573_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X4: real,Y3: real] : ( plus_plus_real @ X4 @ ( uminus_uminus_real @ Y3 ) ) ) ) ).

% minus_real_def
thf(fact_5574_sum__subtractf__nat,axiom,
    ! [A4: set_int,G: int > nat,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5575_sum__subtractf__nat,axiom,
    ! [A4: set_real,G: real > nat,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X4: real] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5576_sum__subtractf__nat,axiom,
    ! [A4: set_complex,G: complex > nat,F: complex > nat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups5693394587270226106ex_nat
          @ ^ [X4: complex] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5577_sum__subtractf__nat,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups977919841031483927at_nat
          @ ^ [X4: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ ( groups977919841031483927at_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5578_sum__subtractf__nat,axiom,
    ! [A4: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_5579_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_5580_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > real,M2: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_5581_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M2: nat,K2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_5582_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > real,M2: nat,K2: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_5583_tanh__real__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_5584_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_5585_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_5586_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
          | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_5587_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_5588_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
          | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_5589_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_5590_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_5591_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_5592_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
        = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_5593_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_5594_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_5595_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
        = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_5596_zero__le__power__abs,axiom,
    ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_5597_zero__le__power__abs,axiom,
    ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_5598_zero__le__power__abs,axiom,
    ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_5599_zero__le__power__abs,axiom,
    ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_5600_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
        & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5601_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_5602_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5603_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_5604_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_5605_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_5606_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_5607_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_5608_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5609_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_5610_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_5611_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_5612_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
        & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5613_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_5614_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5615_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_5616_sum__eq__Suc0__iff,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( ( F @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y3: complex] :
                  ( ( member_complex @ Y3 @ A4 )
                 => ( ( X4 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_5617_sum__eq__Suc0__iff,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A4 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( ( F @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y3: nat] :
                  ( ( member_nat @ Y3 @ A4 )
                 => ( ( X4 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_5618_sum__SucD,axiom,
    ! [F: nat > nat,A4: set_nat,N: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A4 )
        = ( suc @ N ) )
     => ? [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ).

% sum_SucD
thf(fact_5619_sum__eq__1__iff,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
          = one_one_nat )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( ( F @ X4 )
                = one_one_nat )
              & ! [Y3: complex] :
                  ( ( member_complex @ Y3 @ A4 )
                 => ( ( X4 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5620_sum__eq__1__iff,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A4 )
          = one_one_nat )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( ( F @ X4 )
                = one_one_nat )
              & ! [Y3: nat] :
                  ( ( member_nat @ Y3 @ A4 )
                 => ( ( X4 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5621_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).

% neg_numeral_le_zero
thf(fact_5622_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_5623_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).

% neg_numeral_le_zero
thf(fact_5624_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).

% neg_numeral_le_zero
thf(fact_5625_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5626_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5627_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5628_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5629_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5630_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5631_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5632_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5633_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).

% neg_numeral_less_zero
thf(fact_5634_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).

% neg_numeral_less_zero
thf(fact_5635_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).

% neg_numeral_less_zero
thf(fact_5636_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_5637_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_5638_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_5639_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(3)
thf(fact_5640_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_5641_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_5642_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_5643_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% le_minus_one_simps(1)
thf(fact_5644_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_5645_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_5646_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_5647_less__minus__one__simps_I1_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% less_minus_one_simps(1)
thf(fact_5648_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_5649_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_5650_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_5651_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(3)
thf(fact_5652_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_5653_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real ) ).

% neg_numeral_le_one
thf(fact_5654_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_5655_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat ) ).

% neg_numeral_le_one
thf(fact_5656_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).

% neg_numeral_le_one
thf(fact_5657_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_5658_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_5659_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_5660_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_5661_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% neg_numeral_le_neg_one
thf(fact_5662_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% neg_numeral_le_neg_one
thf(fact_5663_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% neg_numeral_le_neg_one
thf(fact_5664_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% neg_numeral_le_neg_one
thf(fact_5665_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_le_neg_one
thf(fact_5666_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_5667_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_le_neg_one
thf(fact_5668_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_le_neg_one
thf(fact_5669_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5670_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5671_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5672_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5673_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5674_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5675_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5676_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5677_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5678_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5679_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5680_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5681_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_less_neg_one
thf(fact_5682_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_less_neg_one
thf(fact_5683_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_less_neg_one
thf(fact_5684_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_5685_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_5686_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_5687_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_5688_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_5689_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real ) ).

% neg_numeral_less_one
thf(fact_5690_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).

% neg_numeral_less_one
thf(fact_5691_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat ) ).

% neg_numeral_less_one
thf(fact_5692_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_5693_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_5694_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_5695_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = ( uminus_uminus_rat @ B ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_5696_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_5697_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_5698_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B )
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_5699_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_5700_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_5701_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
          = C )
        = ( ( uminus_uminus_rat @ A )
          = ( times_times_rat @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_5702_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_5703_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_5704_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
        = ( ( times_times_rat @ C @ B )
          = ( uminus_uminus_rat @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_5705_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5706_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5707_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5708_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5709_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5710_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5711_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5712_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5713_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5714_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5715_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_5716_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_5717_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ( B != zero_zero_rat )
        & ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_5718_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_5719_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_5720_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_5721_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_5722_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_5723_power__minus,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).

% power_minus
thf(fact_5724_power__minus,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).

% power_minus
thf(fact_5725_power__minus,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_minus
thf(fact_5726_power__minus,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_minus
thf(fact_5727_power__minus,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_minus
thf(fact_5728_power__minus__Bit0,axiom,
    ! [X: real,K2: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) )
      = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ) ).

% power_minus_Bit0
thf(fact_5729_power__minus__Bit0,axiom,
    ! [X: int,K2: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) )
      = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ) ).

% power_minus_Bit0
thf(fact_5730_power__minus__Bit0,axiom,
    ! [X: complex,K2: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) )
      = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ) ).

% power_minus_Bit0
thf(fact_5731_power__minus__Bit0,axiom,
    ! [X: rat,K2: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) )
      = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ) ).

% power_minus_Bit0
thf(fact_5732_power__minus__Bit0,axiom,
    ! [X: code_integer,K2: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) )
      = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ) ).

% power_minus_Bit0
thf(fact_5733_lemma__interval__lt,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
               => ( ( ord_less_real @ A @ Y5 )
                  & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_5734_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_less_0_iff
thf(fact_5735_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_less_add_iff
thf(fact_5736_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_le_0_iff
thf(fact_5737_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_le_add_iff
thf(fact_5738_sum__power__add,axiom,
    ! [X: complex,M2: nat,I6: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M2 @ I3 ) )
        @ I6 )
      = ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_5739_sum__power__add,axiom,
    ! [X: rat,M2: nat,I6: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M2 @ I3 ) )
        @ I6 )
      = ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_5740_sum__power__add,axiom,
    ! [X: int,M2: nat,I6: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M2 @ I3 ) )
        @ I6 )
      = ( times_times_int @ ( power_power_int @ X @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_5741_sum__power__add,axiom,
    ! [X: real,M2: nat,I6: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M2 @ I3 ) )
        @ I6 )
      = ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_5742_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).

% zmod_zminus2_eq_if
thf(fact_5743_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% zmod_zminus1_eq_if
thf(fact_5744_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_5745_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N: nat,M2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_5746_sum__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X4: complex] : X4
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_5747_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X4: complex] : X4
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_5748_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5749_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_5750_abs__add__one__gt__zero,axiom,
    ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_5751_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_5752_sum__diff__nat,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B4 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_5753_sum__diff__nat,axiom,
    ! [B4: set_nat,A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B4 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_5754_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > complex,K2: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5755_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > rat,K2: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_rat )
     => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5756_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > int,K2: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5757_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > nat,K2: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5758_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > real,K2: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5759_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5760_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5761_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5762_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5763_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5764_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5765_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5766_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5767_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( plus_plus_rat @ ( G @ M2 ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5768_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( plus_plus_int @ ( G @ M2 ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5769_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( plus_plus_nat @ ( G @ M2 ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5770_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( plus_plus_real @ ( G @ M2 ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5771_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_5772_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_5773_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_5774_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_5775_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_5776_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_5777_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_5778_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_5779_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_5780_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_5781_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_5782_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_5783_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5784_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5785_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5786_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5787_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5788_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5789_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_5790_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_5791_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5792_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_5793_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_5794_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5795_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_5796_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_5797_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5798_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_5799_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_5800_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5801_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_5802_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_5803_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5804_even__minus,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_minus
thf(fact_5805_even__minus,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_minus
thf(fact_5806_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
               => ( ( ord_less_eq_real @ A @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_5807_power2__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5808_power2__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5809_power2__eq__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1482373934393186551omplex @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5810_power2__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5811_power2__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5812_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self2
thf(fact_5813_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G @ M2 )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5814_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G @ M2 )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5815_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G @ M2 )
          @ ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5816_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G @ M2 )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5817_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_5818_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_5819_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( minus_minus_real @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_5820_div__eq__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% div_eq_minus1
thf(fact_5821_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5822_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_5823_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_5824_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_5825_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( abs_abs_Code_integer @ X )
        = one_one_Code_integer ) ) ).

% abs_square_eq_1
thf(fact_5826_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( abs_abs_rat @ X )
        = one_one_rat ) ) ).

% abs_square_eq_1
thf(fact_5827_abs__square__eq__1,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( abs_abs_real @ X )
        = one_one_real ) ) ).

% abs_square_eq_1
thf(fact_5828_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% abs_square_eq_1
thf(fact_5829_power__even__abs,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
        = ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_even_abs
thf(fact_5830_power__even__abs,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
        = ( power_power_real @ A @ N ) ) ) ).

% power_even_abs
thf(fact_5831_power__even__abs,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
        = ( power_power_int @ A @ N ) ) ) ).

% power_even_abs
thf(fact_5832_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > rat,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5833_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > int,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5834_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > nat,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5835_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > real,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5836_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_5837_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_5838_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_5839_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_5840_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_5841_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_5842_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_5843_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_5844_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_5845_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_5846_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_5847_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_5848_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_5849_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_5850_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_5851_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_5852_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( A = one_one_real )
        | ( A
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5853_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( A = one_one_int )
        | ( A
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5854_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
      = ( ( A = one_one_complex )
        | ( A
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5855_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( A = one_one_rat )
        | ( A
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5856_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( A = one_one_Code_integer )
        | ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5857_uminus__power__if,axiom,
    ! [N: nat,A: real] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
          = ( power_power_real @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
          = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5858_uminus__power__if,axiom,
    ! [N: nat,A: int] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
          = ( power_power_int @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
          = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5859_uminus__power__if,axiom,
    ! [N: nat,A: complex] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
          = ( power_power_complex @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5860_uminus__power__if,axiom,
    ! [N: nat,A: rat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
          = ( power_power_rat @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
          = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5861_uminus__power__if,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
          = ( power_8256067586552552935nteger @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5862_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K2 ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5863_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K2 ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5864_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K2 ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5865_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K2 ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5866_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K2 ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5867_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_5868_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_5869_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K2 ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_5870_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ K2 )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_5871_signed__take__bit__int__greater__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_int @ K2 @ ( bit_ri631733984087533419it_int @ N @ K2 ) )
      = ( ord_less_int @ K2 @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_5872_minus__mod__int__eq,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K2 ) @ L )
        = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K2 @ one_one_int ) @ L ) ) ) ) ).

% minus_mod_int_eq
thf(fact_5873_zmod__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( minus_minus_int @ B @ one_one_int ) ) ) ).

% zmod_minus1
thf(fact_5874_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus1_eq_if
thf(fact_5875_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus2_eq_if
thf(fact_5876_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X: code_integer] :
      ( ! [X5: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X5 )
         => ( P @ X5 @ ( power_8256067586552552935nteger @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5877_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
         => ( P @ X5 @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5878_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X5 )
         => ( P @ X5 @ ( power_power_rat @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5879_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X: int] :
      ( ! [X5: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X5 )
         => ( P @ X5 @ ( power_power_int @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_5880_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5881_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_5882_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_5883_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_5884_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_5885_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_5886_abs__square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_5887_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_5888_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_5889_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_5890_abs__square__less__1,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_5891_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_5892_power__mono__even,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_5893_power__mono__even,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_5894_power__mono__even,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_5895_power__mono__even,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_5896_zminus1__lemma,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q3 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q3 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_5897_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_5898_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_5899_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_5900_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_5901_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X @ I6 )
          = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I6 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_5902_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X @ I6 )
          = one_one_Code_integer )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I6 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7873554091576472773nteger
                  @ ^ [I3: int] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_5903_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7713935264441627589nteger @ X @ I6 )
          = one_one_Code_integer )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7713935264441627589nteger
                  @ ^ [I3: real] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_5904_convex__sum__bound__le,axiom,
    ! [I6: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups6621422865394947399nteger @ X @ I6 )
          = one_one_Code_integer )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups6621422865394947399nteger
                  @ ^ [I3: complex] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_5905_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: int > real,A: int > real,B: real,Delta: real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
     => ( ( ( groups8778361861064173332t_real @ X @ I6 )
          = one_one_real )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8778361861064173332t_real
                  @ ^ [I3: int] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_5906_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: real > real,A: real > real,B: real,Delta: real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
     => ( ( ( groups8097168146408367636l_real @ X @ I6 )
          = one_one_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8097168146408367636l_real
                  @ ^ [I3: real] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_5907_convex__sum__bound__le,axiom,
    ! [I6: set_complex,X: complex > real,A: complex > real,B: real,Delta: real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
     => ( ( ( groups5808333547571424918x_real @ X @ I6 )
          = one_one_real )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups5808333547571424918x_real
                  @ ^ [I3: complex] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_5908_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I6 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X @ I6 )
          = one_one_rat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I6 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_5909_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I6 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups3906332499630173760nt_rat @ X @ I6 )
          = one_one_rat )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I6 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups3906332499630173760nt_rat
                  @ ^ [I3: int] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_5910_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I6 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups1300246762558778688al_rat @ X @ I6 )
          = one_one_rat )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups1300246762558778688al_rat
                  @ ^ [I3: real] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_5911_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_5912_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_5913_square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
     => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_5914_square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_5915_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_5916_minus__power__mult__self,axiom,
    ! [A: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_5917_minus__power__mult__self,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_5918_minus__power__mult__self,axiom,
    ! [A: complex,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_5919_minus__power__mult__self,axiom,
    ! [A: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_5920_minus__power__mult__self,axiom,
    ! [A: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_5921_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% minus_one_power_iff
thf(fact_5922_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_5923_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
          = one_one_complex ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% minus_one_power_iff
thf(fact_5924_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_5925_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_5926_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F: nat > complex] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups2073611262835488442omplex
            @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( minus_minus_complex @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups2073611262835488442omplex
            @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
thf(fact_5927_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( minus_minus_rat @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_5928_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( minus_minus_int @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_5929_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( minus_minus_real @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_5930_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
        = ( minus_minus_rat @ ( F @ N ) @ ( F @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_5931_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3539618377306564664at_int
          @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
        = ( minus_minus_int @ ( F @ N ) @ ( F @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_5932_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups6591440286371151544t_real
          @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
        = ( minus_minus_real @ ( F @ N ) @ ( F @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_5933_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ K2 )
        = K2 )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 )
        & ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_5934_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 )
     => ( ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_ri631733984087533419it_int @ N @ K2 )
          = K2 ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_5935_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_exp_eq_int
thf(fact_5936_div__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K2 @ L ) @ zero_zero_int )
       => ( ( divide_divide_int @ K2 @ L )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_5937_mask__eq__sum__exp,axiom,
    ! [N: nat] :
      ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
      = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_5938_mask__eq__sum__exp,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_5939_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: complex] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X @ M2 ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5940_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X @ M2 ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5941_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
        = ( minus_minus_int @ ( power_power_int @ X @ M2 ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5942_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
        = ( minus_minus_real @ ( power_power_real @ X @ M2 ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5943_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_real @ one_one_real ) ) ).

% power_minus1_odd
thf(fact_5944_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_5945_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power_minus1_odd
thf(fact_5946_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_rat @ one_one_rat ) ) ).

% power_minus1_odd
thf(fact_5947_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% power_minus1_odd
thf(fact_5948_sum_Oin__pairs,axiom,
    ! [G: nat > rat,M2: nat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.in_pairs
thf(fact_5949_sum_Oin__pairs,axiom,
    ! [G: nat > int,M2: nat,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.in_pairs
thf(fact_5950_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.in_pairs
thf(fact_5951_sum_Oin__pairs,axiom,
    ! [G: nat > real,M2: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.in_pairs
thf(fact_5952_int__bit__induct,axiom,
    ! [P: int > $o,K2: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K: int] :
              ( ( P @ K )
             => ( ( K != zero_zero_int )
               => ( P @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K: int] :
                ( ( P @ K )
               => ( ( K
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K2 ) ) ) ) ) ).

% int_bit_induct
thf(fact_5953_eq__diff__eq_H,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( X
        = ( minus_minus_real @ Y @ Z ) )
      = ( Y
        = ( plus_plus_real @ X @ Z ) ) ) ).

% eq_diff_eq'
thf(fact_5954_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_5955_gauss__sum__nat,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_5956_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_5957_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_int @ K2 @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K2 ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_5958_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_5959_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_5960_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_5961_Sum__Icc__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_5962_arctan__double,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_5963_divmod__algorithm__code_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5055182867167087721od_nat @ M2 @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_5964_divmod__algorithm__code_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5052692396658037445od_int @ M2 @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_5965_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M3: nat,N3: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N3 = zero_zero_nat )
            | ( ord_less_nat @ M3 @ N3 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_5966_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_5967_sum__gp,axiom,
    ! [N: nat,M2: nat,X: complex] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( ( X = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M2 ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5968_sum__gp,axiom,
    ! [N: nat,M2: nat,X: rat] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( ( X = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M2 ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5969_sum__gp,axiom,
    ! [N: nat,M2: nat,X: real] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( ( X = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M2 ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5970_even__set__encode__iff,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A4 ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A4 ) ) ) ) ).

% even_set_encode_iff
thf(fact_5971_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_5972_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_5973_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M2 )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_5974_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M2 )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_5975_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri681578069525770553at_rat @ M2 )
        = ( semiri681578069525770553at_rat @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_5976_semiring__norm_I90_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( bit1 @ M2 )
        = ( bit1 @ N ) )
      = ( M2 = N ) ) ).

% semiring_norm(90)
thf(fact_5977_case__prodI,axiom,
    ! [F: code_integer > $o > $o,A: code_integer,B: $o] :
      ( ( F @ A @ B )
     => ( produc7828578312038201481er_o_o @ F @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ).

% case_prodI
thf(fact_5978_case__prodI,axiom,
    ! [F: num > num > $o,A: num,B: num] :
      ( ( F @ A @ B )
     => ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) ) ) ).

% case_prodI
thf(fact_5979_case__prodI,axiom,
    ! [F: nat > num > $o,A: nat,B: num] :
      ( ( F @ A @ B )
     => ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) ) ) ).

% case_prodI
thf(fact_5980_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_5981_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_5982_case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,C: code_integer > $o > $o] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P2
            = ( produc6677183202524767010eger_o @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc7828578312038201481er_o_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_5983_case__prodI2,axiom,
    ! [P2: product_prod_num_num,C: num > num > $o] :
      ( ! [A3: num,B3: num] :
          ( ( P2
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc5703948589228662326_num_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_5984_case__prodI2,axiom,
    ! [P2: product_prod_nat_num,C: nat > num > $o] :
      ( ! [A3: nat,B3: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc4927758841916487424_num_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_5985_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc6081775807080527818_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_5986_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C: int > int > $o] :
      ( ! [A3: int,B3: int] :
          ( ( P2
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc4947309494688390418_int_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_5987_mem__case__prodI,axiom,
    ! [Z: nat,C: code_integer > $o > set_nat,A: code_integer,B: $o] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5988_mem__case__prodI,axiom,
    ! [Z: int,C: code_integer > $o > set_int,A: code_integer,B: $o] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5989_mem__case__prodI,axiom,
    ! [Z: real,C: code_integer > $o > set_real,A: code_integer,B: $o] :
      ( ( member_real @ Z @ ( C @ A @ B ) )
     => ( member_real @ Z @ ( produc242741666403216561t_real @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5990_mem__case__prodI,axiom,
    ! [Z: complex,C: code_integer > $o > set_complex,A: code_integer,B: $o] :
      ( ( member_complex @ Z @ ( C @ A @ B ) )
     => ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5991_mem__case__prodI,axiom,
    ! [Z: nat,C: num > num > set_nat,A: num,B: num] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5992_mem__case__prodI,axiom,
    ! [Z: int,C: num > num > set_int,A: num,B: num] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5993_mem__case__prodI,axiom,
    ! [Z: real,C: num > num > set_real,A: num,B: num] :
      ( ( member_real @ Z @ ( C @ A @ B ) )
     => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5994_mem__case__prodI,axiom,
    ! [Z: complex,C: num > num > set_complex,A: num,B: num] :
      ( ( member_complex @ Z @ ( C @ A @ B ) )
     => ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5995_mem__case__prodI,axiom,
    ! [Z: nat,C: nat > num > set_nat,A: nat,B: num] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5996_mem__case__prodI,axiom,
    ! [Z: int,C: nat > num > set_int,A: nat,B: num] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc9175805072616146554et_int @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_5997_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: nat,C: code_integer > $o > set_nat] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P2
            = ( produc6677183202524767010eger_o @ A3 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_5998_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: int,C: code_integer > $o > set_int] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P2
            = ( produc6677183202524767010eger_o @ A3 @ B3 ) )
         => ( member_int @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_5999_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: real,C: code_integer > $o > set_real] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P2
            = ( produc6677183202524767010eger_o @ A3 @ B3 ) )
         => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6000_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: complex,C: code_integer > $o > set_complex] :
      ( ! [A3: code_integer,B3: $o] :
          ( ( P2
            = ( produc6677183202524767010eger_o @ A3 @ B3 ) )
         => ( member_complex @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6001_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: nat,C: num > num > set_nat] :
      ( ! [A3: num,B3: num] :
          ( ( P2
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6002_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: int,C: num > num > set_int] :
      ( ! [A3: num,B3: num] :
          ( ( P2
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( member_int @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6003_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: real,C: num > num > set_real] :
      ( ! [A3: num,B3: num] :
          ( ( P2
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6004_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: complex,C: num > num > set_complex] :
      ( ! [A3: num,B3: num] :
          ( ( P2
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( member_complex @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6005_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: nat,C: nat > num > set_nat] :
      ( ! [A3: nat,B3: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A3 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6006_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: int,C: nat > num > set_int] :
      ( ! [A3: nat,B3: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A3 @ B3 ) )
         => ( member_int @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_int @ Z @ ( produc9175805072616146554et_int @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6007_case__prodI2_H,axiom,
    ! [P2: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ( product_Pair_nat_nat @ A3 @ B3 )
            = P2 )
         => ( C @ A3 @ B3 @ X ) )
     => ( produc8739625826339149834_nat_o @ C @ P2 @ X ) ) ).

% case_prodI2'
thf(fact_6008_semiring__norm_I88_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit0 @ M2 )
     != ( bit1 @ N ) ) ).

% semiring_norm(88)
thf(fact_6009_semiring__norm_I89_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit1 @ M2 )
     != ( bit0 @ N ) ) ).

% semiring_norm(89)
thf(fact_6010_semiring__norm_I84_J,axiom,
    ! [N: num] :
      ( one
     != ( bit1 @ N ) ) ).

% semiring_norm(84)
thf(fact_6011_semiring__norm_I86_J,axiom,
    ! [M2: num] :
      ( ( bit1 @ M2 )
     != one ) ).

% semiring_norm(86)
thf(fact_6012_int__eq__iff__numeral,axiom,
    ! [M2: nat,V: num] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = ( numeral_numeral_int @ V ) )
      = ( M2
        = ( numeral_numeral_nat @ V ) ) ) ).

% int_eq_iff_numeral
thf(fact_6013_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% abs_of_nat
thf(fact_6014_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_6015_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% abs_of_nat
thf(fact_6016_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% abs_of_nat
thf(fact_6017_zdvd1__eq,axiom,
    ! [X: int] :
      ( ( dvd_dvd_int @ X @ one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_6018_semiring__norm_I80_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% semiring_norm(80)
thf(fact_6019_semiring__norm_I73_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% semiring_norm(73)
thf(fact_6020_set__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_set_encode @ ( nat_set_decode @ N ) )
      = N ) ).

% set_decode_inverse
thf(fact_6021_of__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% of_nat_0
thf(fact_6022_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_6023_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_6024_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_6025_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_6026_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_6027_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_6028_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_6029_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_6030_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_6031_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri8010041392384452111omplex @ M2 )
        = zero_zero_complex )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_6032_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = zero_zero_int )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_6033_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M2 )
        = zero_zero_real )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_6034_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M2 )
        = zero_zero_nat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_6035_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri681578069525770553at_rat @ M2 )
        = zero_zero_rat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_6036_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_6037_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_6038_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_6039_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_6040_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_6041_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_6042_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_6043_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_6044_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
      = ( numera6690914467698888265omplex @ N ) ) ).

% of_nat_numeral
thf(fact_6045_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% of_nat_numeral
thf(fact_6046_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% of_nat_numeral
thf(fact_6047_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% of_nat_numeral
thf(fact_6048_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% of_nat_numeral
thf(fact_6049_of__nat__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_6050_of__nat__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_add
thf(fact_6051_of__nat__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_6052_of__nat__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_add
thf(fact_6053_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_6054_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_mult
thf(fact_6055_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_6056_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_mult
thf(fact_6057_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_6058_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_6059_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_6060_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_6061_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_6062_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_6063_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_6064_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_6065_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_6066_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_6067_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri8010041392384452111omplex @ N )
        = one_one_complex )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_6068_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1314217659103216013at_int @ N )
        = one_one_int )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_6069_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri5074537144036343181t_real @ N )
        = one_one_real )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_6070_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N )
        = one_one_nat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_6071_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri681578069525770553at_rat @ N )
        = one_one_rat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_6072_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri8010041392384452111omplex @ X )
        = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6073_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6074_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6075_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6076_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri681578069525770553at_rat @ X )
        = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6077_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
        = ( semiri8010041392384452111omplex @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6078_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
        = ( semiri1314217659103216013at_int @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6079_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
        = ( semiri5074537144036343181t_real @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6080_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
        = ( semiri1316708129612266289at_nat @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6081_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
        = ( semiri681578069525770553at_rat @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6082_of__nat__power,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M2 @ N ) )
      = ( power_power_complex @ ( semiri8010041392384452111omplex @ M2 ) @ N ) ) ).

% of_nat_power
thf(fact_6083_of__nat__power,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M2 @ N ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M2 ) @ N ) ) ).

% of_nat_power
thf(fact_6084_of__nat__power,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M2 @ N ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M2 ) @ N ) ) ).

% of_nat_power
thf(fact_6085_of__nat__power,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M2 @ N ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ N ) ) ).

% of_nat_power
thf(fact_6086_of__nat__power,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M2 @ N ) )
      = ( power_power_rat @ ( semiri681578069525770553at_rat @ M2 ) @ N ) ) ).

% of_nat_power
thf(fact_6087_negative__zless,axiom,
    ! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).

% negative_zless
thf(fact_6088_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_6089_semiring__norm_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M2 @ N ) ) ) ).

% semiring_norm(9)
thf(fact_6090_semiring__norm_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M2 @ N ) ) ) ).

% semiring_norm(7)
thf(fact_6091_semiring__norm_I15_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M2 ) @ N ) ) ) ).

% semiring_norm(15)
thf(fact_6092_semiring__norm_I14_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( bit0 @ ( times_times_num @ M2 @ ( bit1 @ N ) ) ) ) ).

% semiring_norm(14)
thf(fact_6093_semiring__norm_I72_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% semiring_norm(72)
thf(fact_6094_semiring__norm_I81_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% semiring_norm(81)
thf(fact_6095_semiring__norm_I70_J,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M2 ) @ one ) ).

% semiring_norm(70)
thf(fact_6096_semiring__norm_I77_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).

% semiring_norm(77)
thf(fact_6097_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_6098_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% of_nat_of_bool
thf(fact_6099_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_6100_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_6101_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_nat_of_bool
thf(fact_6102_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_6103_set__encode__inverse,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( nat_set_decode @ ( nat_set_encode @ A4 ) )
        = A4 ) ) ).

% set_encode_inverse
thf(fact_6104_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_6105_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M2 ) @ zero_zero_rat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_6106_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_6107_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_6108_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M2 ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_6109_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_6110_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M2 ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_6111_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_6112_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M2 ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_6113_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_6114_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
      = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_6115_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M2: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M2 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M2 ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_6116_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit1
thf(fact_6117_semiring__norm_I3_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N ) )
      = ( bit1 @ N ) ) ).

% semiring_norm(3)
thf(fact_6118_semiring__norm_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).

% semiring_norm(4)
thf(fact_6119_semiring__norm_I5_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_num @ ( bit0 @ M2 ) @ one )
      = ( bit1 @ M2 ) ) ).

% semiring_norm(5)
thf(fact_6120_semiring__norm_I8_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_num @ ( bit1 @ M2 ) @ one )
      = ( bit0 @ ( plus_plus_num @ M2 @ one ) ) ) ).

% semiring_norm(8)
thf(fact_6121_semiring__norm_I10_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M2 @ N ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_6122_semiring__norm_I16_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M2 @ N ) @ ( bit0 @ ( times_times_num @ M2 @ N ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_6123_semiring__norm_I74_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% semiring_norm(74)
thf(fact_6124_semiring__norm_I79_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% semiring_norm(79)
thf(fact_6125_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_6126_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_6127_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_6128_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_6129_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_6130_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_6131_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_6132_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_6133_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_6134_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_6135_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_6136_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_6137_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_6138_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_6139_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_6140_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_6141_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_6142_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_6143_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_6144_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_6145_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri8010041392384452111omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6146_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6147_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6148_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6149_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri681578069525770553at_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6150_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
        = ( semiri8010041392384452111omplex @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_6151_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = ( semiri1314217659103216013at_int @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_6152_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
        = ( semiri5074537144036343181t_real @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_6153_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = ( semiri1316708129612266289at_nat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_6154_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
        = ( semiri681578069525770553at_rat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_6155_dvd__numeral__simp,axiom,
    ! [M2: num,N: num] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) )
      = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M2 ) ) ) ).

% dvd_numeral_simp
thf(fact_6156_dvd__numeral__simp,axiom,
    ! [M2: num,N: num] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M2 ) ) ) ).

% dvd_numeral_simp
thf(fact_6157_dvd__numeral__simp,axiom,
    ! [M2: num,N: num] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M2 ) ) ) ).

% dvd_numeral_simp
thf(fact_6158_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique5055182867167087721od_nat @ M2 @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M2 ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_6159_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique5052692396658037445od_int @ M2 @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M2 ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_6160_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_6161_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_6162_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_6163_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_6164_Suc__div__eq__add3__div__numeral,axiom,
    ! [M2: nat,V: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_6165_div__Suc__eq__div__add3,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide_nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( divide_divide_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_6166_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_6167_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_6168_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M2: nat,V: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_6169_mod__Suc__eq__mod__add3,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( modulo_modulo_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_6170_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_6171_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_6172_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_6173_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_6174_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_6175_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_6176_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_6177_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_6178_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_6179_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_6180_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_6181_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_6182_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_6183_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_6184_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_6185_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_6186_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_6187_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_6188_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_6189_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_6190_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_6191_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).

% zmod_numeral_Bit1
thf(fact_6192_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_num @ M2 @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_6193_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_num @ M2 @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_6194_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_num @ M2 @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_6195_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_num @ M2 @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_6196_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_num @ M2 @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_6197_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_num @ M2 @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_6198_divmod__algorithm__code_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M2 @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6199_divmod__algorithm__code_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M2 @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6200_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_6201_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_6202_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_6203_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_6204_mem__case__prodE,axiom,
    ! [Z: nat,C: code_integer > $o > set_nat,P2: produc6271795597528267376eger_o] :
      ( ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P2 ) )
     => ~ ! [X5: code_integer,Y4: $o] :
            ( ( P2
              = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
           => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6205_mem__case__prodE,axiom,
    ! [Z: int,C: code_integer > $o > set_int,P2: produc6271795597528267376eger_o] :
      ( ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P2 ) )
     => ~ ! [X5: code_integer,Y4: $o] :
            ( ( P2
              = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
           => ~ ( member_int @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6206_mem__case__prodE,axiom,
    ! [Z: real,C: code_integer > $o > set_real,P2: produc6271795597528267376eger_o] :
      ( ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P2 ) )
     => ~ ! [X5: code_integer,Y4: $o] :
            ( ( P2
              = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
           => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6207_mem__case__prodE,axiom,
    ! [Z: complex,C: code_integer > $o > set_complex,P2: produc6271795597528267376eger_o] :
      ( ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P2 ) )
     => ~ ! [X5: code_integer,Y4: $o] :
            ( ( P2
              = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
           => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6208_mem__case__prodE,axiom,
    ! [Z: nat,C: num > num > set_nat,P2: product_prod_num_num] :
      ( ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P2 ) )
     => ~ ! [X5: num,Y4: num] :
            ( ( P2
              = ( product_Pair_num_num @ X5 @ Y4 ) )
           => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6209_mem__case__prodE,axiom,
    ! [Z: int,C: num > num > set_int,P2: product_prod_num_num] :
      ( ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P2 ) )
     => ~ ! [X5: num,Y4: num] :
            ( ( P2
              = ( product_Pair_num_num @ X5 @ Y4 ) )
           => ~ ( member_int @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6210_mem__case__prodE,axiom,
    ! [Z: real,C: num > num > set_real,P2: product_prod_num_num] :
      ( ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P2 ) )
     => ~ ! [X5: num,Y4: num] :
            ( ( P2
              = ( product_Pair_num_num @ X5 @ Y4 ) )
           => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6211_mem__case__prodE,axiom,
    ! [Z: complex,C: num > num > set_complex,P2: product_prod_num_num] :
      ( ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P2 ) )
     => ~ ! [X5: num,Y4: num] :
            ( ( P2
              = ( product_Pair_num_num @ X5 @ Y4 ) )
           => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6212_mem__case__prodE,axiom,
    ! [Z: nat,C: nat > num > set_nat,P2: product_prod_nat_num] :
      ( ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ P2 ) )
     => ~ ! [X5: nat,Y4: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X5 @ Y4 ) )
           => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6213_mem__case__prodE,axiom,
    ! [Z: int,C: nat > num > set_int,P2: product_prod_nat_num] :
      ( ( member_int @ Z @ ( produc9175805072616146554et_int @ C @ P2 ) )
     => ~ ! [X5: nat,Y4: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X5 @ Y4 ) )
           => ~ ( member_int @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6214_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] :
      ( ( bit0 @ X2 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_6215_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_6216_int__of__nat__induct,axiom,
    ! [P: int > $o,Z: int] :
      ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
       => ( P @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_6217_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% int_cases
thf(fact_6218_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M: nat,N2: nat] :
          ( Z
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% int_diff_cases
thf(fact_6219_case__prodD,axiom,
    ! [F: code_integer > $o > $o,A: code_integer,B: $o] :
      ( ( produc7828578312038201481er_o_o @ F @ ( produc6677183202524767010eger_o @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_6220_case__prodD,axiom,
    ! [F: num > num > $o,A: num,B: num] :
      ( ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_6221_case__prodD,axiom,
    ! [F: nat > num > $o,A: nat,B: num] :
      ( ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_6222_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_6223_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_6224_case__prodE,axiom,
    ! [C: code_integer > $o > $o,P2: produc6271795597528267376eger_o] :
      ( ( produc7828578312038201481er_o_o @ C @ P2 )
     => ~ ! [X5: code_integer,Y4: $o] :
            ( ( P2
              = ( produc6677183202524767010eger_o @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6225_case__prodE,axiom,
    ! [C: num > num > $o,P2: product_prod_num_num] :
      ( ( produc5703948589228662326_num_o @ C @ P2 )
     => ~ ! [X5: num,Y4: num] :
            ( ( P2
              = ( product_Pair_num_num @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6226_case__prodE,axiom,
    ! [C: nat > num > $o,P2: product_prod_nat_num] :
      ( ( produc4927758841916487424_num_o @ C @ P2 )
     => ~ ! [X5: nat,Y4: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6227_case__prodE,axiom,
    ! [C: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P2 )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6228_case__prodE,axiom,
    ! [C: int > int > $o,P2: product_prod_int_int] :
      ( ( produc4947309494688390418_int_o @ C @ P2 )
     => ~ ! [X5: int,Y4: int] :
            ( ( P2
              = ( product_Pair_int_int @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6229_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_6230_case__prodE_H,axiom,
    ! [C: nat > nat > product_prod_nat_nat > $o,P2: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ C @ P2 @ Z )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 @ Z ) ) ) ).

% case_prodE'
thf(fact_6231_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).

% of_nat_0_le_iff
thf(fact_6232_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_6233_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_6234_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).

% of_nat_0_le_iff
thf(fact_6235_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_6236_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_6237_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_6238_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_6239_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
     != zero_zero_complex ) ).

% of_nat_neq_0
thf(fact_6240_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_6241_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_6242_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_6243_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_6244_div__mult2__eq_H,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% div_mult2_eq'
thf(fact_6245_div__mult2__eq_H,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% div_mult2_eq'
thf(fact_6246_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_6247_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_6248_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_6249_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_6250_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_6251_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_6252_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_6253_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_6254_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_6255_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).

% of_nat_mono
thf(fact_6256_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_6257_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_6258_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N2: num] :
            ( X
           != ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
       => ( ! [N2: num] :
              ( X
             != ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
         => ( ! [M: num] :
                ( X
               != ( product_Pair_num_num @ ( bit0 @ M ) @ one ) )
           => ( ! [M: num,N2: num] :
                  ( X
                 != ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) ) )
             => ( ! [M: num,N2: num] :
                    ( X
                   != ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) )
               => ( ! [M: num] :
                      ( X
                     != ( product_Pair_num_num @ ( bit1 @ M ) @ one ) )
                 => ( ! [M: num,N2: num] :
                        ( X
                       != ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) )
                   => ~ ! [M: num,N2: num] :
                          ( X
                         != ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_6259_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X22: num] :
            ( Y
           != ( bit0 @ X22 ) )
       => ~ ! [X33: num] :
              ( Y
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_6260_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_6261_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M2 @ N ) )
      = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_6262_int__cases4,axiom,
    ! [M2: int] :
      ( ! [N2: nat] :
          ( M2
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( M2
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% int_cases4
thf(fact_6263_of__nat__dvd__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( dvd_dvd_nat @ M2 @ N ) ) ).

% of_nat_dvd_iff
thf(fact_6264_of__nat__dvd__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( dvd_dvd_nat @ M2 @ N ) ) ).

% of_nat_dvd_iff
thf(fact_6265_of__nat__dvd__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( dvd_dvd_nat @ M2 @ N ) ) ).

% of_nat_dvd_iff
thf(fact_6266_int__ops_I3_J,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% int_ops(3)
thf(fact_6267_abs__zmult__eq__1,axiom,
    ! [M2: int,N: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M2 @ N ) )
        = one_one_int )
     => ( ( abs_abs_int @ M2 )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_6268_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_6269_zle__int,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% zle_int
thf(fact_6270_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_6271_of__nat__mod,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mod
thf(fact_6272_of__nat__mod,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M2 @ N ) )
      = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mod
thf(fact_6273_set__encode__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( ( nat_set_encode @ A4 )
            = ( nat_set_encode @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% set_encode_eq
thf(fact_6274_zadd__int__left,axiom,
    ! [M2: nat,N: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_6275_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_6276_int__plus,axiom,
    ! [N: nat,M2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).

% int_plus
thf(fact_6277_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_6278_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(7)
thf(fact_6279_zdiv__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zdiv_int
thf(fact_6280_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_6281_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_6282_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_6283_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_6284_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X ) @ ( semiri681578069525770553at_rat @ Y ) ) ) ).

% of_nat_max
thf(fact_6285_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).

% nat_less_as_int
thf(fact_6286_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).

% nat_leq_as_int
thf(fact_6287_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).

% numeral_Bit1
thf(fact_6288_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).

% numeral_Bit1
thf(fact_6289_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_6290_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_6291_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_6292_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% of_nat_diff
thf(fact_6293_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M2 @ N ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% of_nat_diff
thf(fact_6294_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_6295_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M2 @ N ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_6296_eval__nat__numeral_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_6297_cong__exp__iff__simps_I10_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6298_cong__exp__iff__simps_I10_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6299_cong__exp__iff__simps_I12_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6300_cong__exp__iff__simps_I12_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6301_cong__exp__iff__simps_I13_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ Q3 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6302_cong__exp__iff__simps_I13_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ Q3 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6303_power__minus__Bit1,axiom,
    ! [X: real,K2: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6304_power__minus__Bit1,axiom,
    ! [X: int,K2: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6305_power__minus__Bit1,axiom,
    ! [X: complex,K2: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6306_power__minus__Bit1,axiom,
    ! [X: rat,K2: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6307_power__minus__Bit1,axiom,
    ! [X: code_integer,K2: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6308_int__cases3,axiom,
    ! [K2: int] :
      ( ( K2 != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K2
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
       => ~ ! [N2: nat] :
              ( ( K2
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% int_cases3
thf(fact_6309_reals__Archimedean3,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ! [Y5: real] :
        ? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).

% reals_Archimedean3
thf(fact_6310_not__zle__0__negative,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).

% not_zle_0_negative
thf(fact_6311_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_6312_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N2: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% negD
thf(fact_6313_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% real_of_nat_div4
thf(fact_6314_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_6315_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_6316_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W2: int,Z2: int] :
        ? [N3: nat] :
          ( Z2
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_6317_abs__mod__less,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K2 @ L ) ) @ ( abs_abs_int @ L ) ) ) ).

% abs_mod_less
thf(fact_6318_real__of__nat__div,axiom,
    ! [D: nat,N: nat] :
      ( ( dvd_dvd_nat @ D @ N )
     => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
        = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div
thf(fact_6319_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).

% numeral_code(3)
thf(fact_6320_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).

% numeral_code(3)
thf(fact_6321_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_6322_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_6323_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_6324_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6325_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6326_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6327_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6328_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6329_set__encode__inf,axiom,
    ! [A4: set_nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( nat_set_encode @ A4 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_6330_mod__mult2__eq_H,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_6331_mod__mult2__eq_H,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_6332_numeral__Bit1__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N ) ) ).

% numeral_Bit1_div_2
thf(fact_6333_numeral__Bit1__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% numeral_Bit1_div_2
thf(fact_6334_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_6335_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_6336_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_6337_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_6338_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_6339_power3__eq__cube,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_6340_power3__eq__cube,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_6341_power3__eq__cube,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_6342_power3__eq__cube,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_6343_power3__eq__cube,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_6344_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_6345_Suc3__eq__add__3,axiom,
    ! [N: nat] :
      ( ( suc @ ( suc @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).

% Suc3_eq_add_3
thf(fact_6346_field__char__0__class_Oof__nat__div,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M2 @ N ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M2 @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_6347_field__char__0__class_Oof__nat__div,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M2 @ N ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M2 @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_6348_field__char__0__class_Oof__nat__div,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M2 @ N ) )
      = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M2 @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_6349_zero__less__imp__eq__int,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( K2
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_6350_pos__int__cases,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ~ ! [N2: nat] :
            ( ( K2
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% pos_int_cases
thf(fact_6351_neg__int__cases,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ K2 @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( ( K2
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% neg_int_cases
thf(fact_6352_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).

% nat_less_real_le
thf(fact_6353_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N3: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_6354_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K2: nat] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_6355_zdvd__mult__cancel1,axiom,
    ! [M2: int,N: int] :
      ( ( M2 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M2 @ N ) @ M2 )
        = ( ( abs_abs_int @ N )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_6356_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).

% int_ops(6)
thf(fact_6357_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_6358_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_6359_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_6360_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_6361_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_6362_inverse__of__nat__le,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_6363_inverse__of__nat__le,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_6364_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6365_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6366_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6367_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6368_even__abs__add__iff,axiom,
    ! [K2: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K2 ) @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).

% even_abs_add_iff
thf(fact_6369_even__add__abs__iff,axiom,
    ! [K2: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ ( abs_abs_int @ L ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).

% even_add_abs_iff
thf(fact_6370_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 )
       => ( ! [M: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ X ) @ C ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_6371_Suc__div__eq__add3__div,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ N ) ) ).

% Suc_div_eq_add3_div
thf(fact_6372_Suc__mod__eq__add3__mod,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ N ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_6373_zdiff__int__split,axiom,
    ! [P: int > $o,X: nat,Y: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_6374_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).

% real_of_nat_div2
thf(fact_6375_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_6376_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).

% ln_realpow
thf(fact_6377_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_6378_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_6379_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_6380_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F: nat > int,K2: int] :
      ( ! [I2: nat] :
          ( ( ( ord_less_eq_nat @ M2 @ I2 )
            & ( ord_less_nat @ I2 @ N ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( ord_less_eq_int @ ( F @ M2 ) @ K2 )
         => ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
           => ? [I2: nat] :
                ( ( ord_less_eq_nat @ M2 @ I2 )
                & ( ord_less_eq_nat @ I2 @ N )
                & ( ( F @ I2 )
                  = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_6381_mod__exhaust__less__4,axiom,
    ! [M2: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = zero_zero_nat )
      | ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = one_one_nat )
      | ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      | ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).

% mod_exhaust_less_4
thf(fact_6382_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_6383_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).

% decr_lemma
thf(fact_6384_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).

% linear_plus_1_le_power
thf(fact_6385_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality
thf(fact_6386_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_6387_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M3: nat,N3: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N3 ) @ ( modulo_modulo_nat @ M3 @ N3 ) ) ) ) ).

% divmod_nat_def
thf(fact_6388_nat__ivt__aux,axiom,
    ! [N: nat,F: nat > int,K2: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
       => ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N )
              & ( ( F @ I2 )
                = K2 ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_6389_nat0__intermed__int__val,axiom,
    ! [N: nat,F: nat > int,K2: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
       => ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N )
              & ( ( F @ I2 )
                = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_6390_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).

% double_gauss_sum
thf(fact_6391_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).

% double_gauss_sum
thf(fact_6392_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).

% double_gauss_sum
thf(fact_6393_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).

% double_gauss_sum
thf(fact_6394_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).

% double_gauss_sum
thf(fact_6395_double__arith__series,axiom,
    ! [A: complex,D: complex,N: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6396_double__arith__series,axiom,
    ! [A: int,D: int,N: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6397_double__arith__series,axiom,
    ! [A: rat,D: rat,N: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6398_double__arith__series,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6399_double__arith__series,axiom,
    ! [A: real,D: real,N: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_6400_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_6401_gauss__sum,axiom,
    ! [N: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_6402_gauss__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_6403_arith__series,axiom,
    ! [A: int,D: int,N: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_6404_arith__series,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_6405_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6406_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6407_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6408_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6409_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6410_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality_even
thf(fact_6411_sum__gp__offset,axiom,
    ! [X: complex,M2: nat,N: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_6412_sum__gp__offset,axiom,
    ! [X: rat,M2: nat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_6413_sum__gp__offset,axiom,
    ! [X: real,M2: nat,N: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_6414_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N3: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M3 @ N3 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M3 ) ) @ ( unique5026877609467782581ep_nat @ N3 @ ( unique5055182867167087721od_nat @ M3 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6415_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N3: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M3 @ N3 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M3 ) ) @ ( unique5024387138958732305ep_int @ N3 @ ( unique5052692396658037445od_int @ M3 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6416_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M3: num,N3: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M3 @ N3 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M3 ) ) @ ( unique4921790084139445826nteger @ N3 @ ( unique3479559517661332726nteger @ M3 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6417_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
     => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_6418_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_6419_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_6420_of__nat__code__if,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N3: nat] :
          ( if_complex @ ( N3 = zero_zero_nat ) @ zero_zero_complex
          @ ( produc1917071388513777916omplex
            @ ^ [M3: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ one_one_complex ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_6421_of__nat__code__if,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N3: nat] :
          ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int
          @ ( produc6840382203811409530at_int
            @ ^ [M3: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ one_one_int ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_6422_of__nat__code__if,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N3: nat] :
          ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
          @ ( produc1703576794950452218t_real
            @ ^ [M3: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ one_one_real ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_6423_of__nat__code__if,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N3: nat] :
          ( if_rat @ ( N3 = zero_zero_nat ) @ zero_zero_rat
          @ ( produc6207742614233964070at_rat
            @ ^ [M3: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ one_one_rat ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_6424_of__nat__code__if,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N3: nat] :
          ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
          @ ( produc6842872674320459806at_nat
            @ ^ [M3: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ one_one_nat ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_6425_nat__approx__posE,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ~ ! [N2: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_6426_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N2: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_6427_one__div__minus__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% one_div_minus_numeral
thf(fact_6428_minus__one__div__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_div_numeral
thf(fact_6429_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_6430_lemma__termdiff3,axiom,
    ! [H2: real,Z: real,K5: real,N: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_6431_lemma__termdiff3,axiom,
    ! [H2: complex,Z: complex,K5: real,N: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_6432_split__part,axiom,
    ! [P: $o,Q: int > int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [A5: int,B5: int] :
            ( P
            & ( Q @ A5 @ B5 ) ) )
      = ( ^ [Ab: product_prod_int_int] :
            ( P
            & ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_6433_Collect__case__prod__mono,axiom,
    ! [A4: int > int > $o,B4: int > int > $o] :
      ( ( ord_le6741204236512500942_int_o @ A4 @ B4 )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A4 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B4 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_6434_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] :
      ( produc4947309494688390418_int_o
      @ ^ [Uu3: int,Uv3: int] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_6435_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).

% complex_mod_triangle_ineq2
thf(fact_6436_monoseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).

% monoseq_realpow
thf(fact_6437_real__arch__simple,axiom,
    ! [X: real] :
    ? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% real_arch_simple
thf(fact_6438_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N2: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% real_arch_simple
thf(fact_6439_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% reals_Archimedean2
thf(fact_6440_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N2: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% reals_Archimedean2
thf(fact_6441_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N2: nat] :
            ( ~ ( P @ N2 )
            & ( P @ ( suc @ N2 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_6442_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_6443_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N2: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_6444_norm__divide__numeral,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_6445_norm__divide__numeral,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_6446_norm__mult__numeral2,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_6447_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_6448_norm__mult__numeral1,axiom,
    ! [W: num,A: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_6449_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_6450_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_6451_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_6452_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_6453_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_6454_norm__one,axiom,
    ( ( real_V7735802525324610683m_real @ one_one_real )
    = one_one_real ) ).

% norm_one
thf(fact_6455_norm__one,axiom,
    ( ( real_V1022390504157884413omplex @ one_one_complex )
    = one_one_real ) ).

% norm_one
thf(fact_6456_norm__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
      = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).

% norm_minus_commute
thf(fact_6457_norm__minus__commute,axiom,
    ! [A: complex,B: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
      = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).

% norm_minus_commute
thf(fact_6458_norm__mult,axiom,
    ! [X: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult
thf(fact_6459_norm__mult,axiom,
    ! [X: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult
thf(fact_6460_norm__divide,axiom,
    ! [A: real,B: real] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).

% norm_divide
thf(fact_6461_norm__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).

% norm_divide
thf(fact_6462_norm__power,axiom,
    ! [X: real,N: nat] :
      ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) )
      = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).

% norm_power
thf(fact_6463_norm__power,axiom,
    ! [X: complex,N: nat] :
      ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).

% norm_power
thf(fact_6464_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_6465_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_6466_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_6467_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_6468_power__eq__imp__eq__norm,axiom,
    ! [W: real,N: nat,Z: real] :
      ( ( ( power_power_real @ W @ N )
        = ( power_power_real @ Z @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V7735802525324610683m_real @ W )
          = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_6469_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N: nat,Z: complex] :
      ( ( ( power_power_complex @ W @ N )
        = ( power_power_complex @ Z @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V1022390504157884413omplex @ W )
          = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_6470_norm__mult__less,axiom,
    ! [X: real,R2: real,Y: real,S2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S2 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).

% norm_mult_less
thf(fact_6471_norm__mult__less,axiom,
    ! [X: complex,R2: real,Y: complex,S2: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S2 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).

% norm_mult_less
thf(fact_6472_norm__mult__ineq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_6473_norm__mult__ineq,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_6474_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_6475_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_6476_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S2 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).

% norm_add_less
thf(fact_6477_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S2: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S2 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).

% norm_add_less
thf(fact_6478_norm__power__ineq,axiom,
    ! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).

% norm_power_ineq
thf(fact_6479_norm__power__ineq,axiom,
    ! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).

% norm_power_ineq
thf(fact_6480_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S2: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S2 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).

% norm_triangle_mono
thf(fact_6481_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S2: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S2 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).

% norm_triangle_mono
thf(fact_6482_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_6483_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_6484_norm__triangle__le,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_6485_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_6486_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_6487_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_6488_norm__diff__triangle__less,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_6489_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_6490_norm__triangle__sub,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_6491_norm__triangle__sub,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_6492_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).

% norm_triangle_ineq4
thf(fact_6493_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).

% norm_triangle_ineq4
thf(fact_6494_norm__diff__triangle__le,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_6495_norm__diff__triangle__le,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_6496_norm__triangle__le__diff,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_6497_norm__triangle__le__diff,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_6498_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_6499_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_6500_norm__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).

% norm_triangle_ineq2
thf(fact_6501_norm__triangle__ineq2,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% norm_triangle_ineq2
thf(fact_6502_power__eq__1__iff,axiom,
    ! [W: real,N: nat] :
      ( ( ( power_power_real @ W @ N )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W )
          = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_6503_power__eq__1__iff,axiom,
    ! [W: complex,N: nat] :
      ( ( ( power_power_complex @ W @ N )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W )
          = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_6504_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_6505_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_6506_norm__triangle__ineq3,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).

% norm_triangle_ineq3
thf(fact_6507_norm__triangle__ineq3,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% norm_triangle_ineq3
thf(fact_6508_square__norm__one,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
     => ( ( real_V7735802525324610683m_real @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_6509_square__norm__one,axiom,
    ! [X: complex] :
      ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
     => ( ( real_V1022390504157884413omplex @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_6510_norm__power__diff,axiom,
    ! [Z: real,W: real,M2: nat] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M2 ) @ ( power_power_real @ W @ M2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_6511_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M2: nat] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M2 ) @ ( power_power_complex @ W @ M2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_6512_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_minus_bit1
thf(fact_6513_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
            @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N3 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_6514_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6515_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6516_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6517_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6518_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6519_arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arctan @ X )
        = ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_6520_lemma__termdiff2,axiom,
    ! [H2: complex,Z: complex,N: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_complex @ H2
          @ ( groups2073611262835488442omplex
            @ ^ [P5: nat] :
                ( groups2073611262835488442omplex
                @ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q4 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_6521_lemma__termdiff2,axiom,
    ! [H2: rat,Z: rat,N: nat] :
      ( ( H2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_rat @ H2
          @ ( groups2906978787729119204at_rat
            @ ^ [P5: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q4 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_6522_lemma__termdiff2,axiom,
    ! [H2: real,Z: real,N: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_real @ H2
          @ ( groups6591440286371151544t_real
            @ ^ [P5: nat] :
                ( groups6591440286371151544t_real
                @ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q4 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_6523_summable__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( summable_real
        @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_6524_lessThan__iff,axiom,
    ! [I: rat,K2: rat] :
      ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K2 ) )
      = ( ord_less_rat @ I @ K2 ) ) ).

% lessThan_iff
thf(fact_6525_lessThan__iff,axiom,
    ! [I: num,K2: num] :
      ( ( member_num @ I @ ( set_ord_lessThan_num @ K2 ) )
      = ( ord_less_num @ I @ K2 ) ) ).

% lessThan_iff
thf(fact_6526_lessThan__iff,axiom,
    ! [I: int,K2: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K2 ) )
      = ( ord_less_int @ I @ K2 ) ) ).

% lessThan_iff
thf(fact_6527_lessThan__iff,axiom,
    ! [I: nat,K2: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K2 ) )
      = ( ord_less_nat @ I @ K2 ) ) ).

% lessThan_iff
thf(fact_6528_lessThan__iff,axiom,
    ! [I: real,K2: real] :
      ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K2 ) )
      = ( ord_less_real @ I @ K2 ) ) ).

% lessThan_iff
thf(fact_6529_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ one_one_complex )
    = one_one_complex ) ).

% dbl_dec_simps(3)
thf(fact_6530_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_6531_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_6532_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_6533_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6534_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6535_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_6536_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_6537_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_6538_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_6539_eq__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K2 )
        = ( suc @ N ) )
      = ( ( pred_numeral @ K2 )
        = N ) ) ).

% eq_numeral_Suc
thf(fact_6540_Suc__eq__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( ( suc @ N )
        = ( numeral_numeral_nat @ K2 ) )
      = ( N
        = ( pred_numeral @ K2 ) ) ) ).

% Suc_eq_numeral
thf(fact_6541_sum_OlessThan__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_6542_sum_OlessThan__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_6543_sum_OlessThan__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_6544_sum_OlessThan__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_6545_less__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
      = ( ord_less_nat @ ( pred_numeral @ K2 ) @ N ) ) ).

% less_numeral_Suc
thf(fact_6546_less__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
      = ( ord_less_nat @ N @ ( pred_numeral @ K2 ) ) ) ).

% less_Suc_numeral
thf(fact_6547_pred__numeral__simps_I3_J,axiom,
    ! [K2: num] :
      ( ( pred_numeral @ ( bit1 @ K2 ) )
      = ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ).

% pred_numeral_simps(3)
thf(fact_6548_le__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
      = ( ord_less_eq_nat @ N @ ( pred_numeral @ K2 ) ) ) ).

% le_Suc_numeral
thf(fact_6549_le__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K2 ) @ N ) ) ).

% le_numeral_Suc
thf(fact_6550_diff__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
      = ( minus_minus_nat @ ( pred_numeral @ K2 ) @ N ) ) ).

% diff_numeral_Suc
thf(fact_6551_diff__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
      = ( minus_minus_nat @ N @ ( pred_numeral @ K2 ) ) ) ).

% diff_Suc_numeral
thf(fact_6552_max__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K2 ) @ N ) ) ) ).

% max_numeral_Suc
thf(fact_6553_max__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
      = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K2 ) ) ) ) ).

% max_Suc_numeral
thf(fact_6554_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_6555_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_6556_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_6557_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_6558_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_6559_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_6560_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_6561_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_bit0
thf(fact_6562_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_minus_bit0
thf(fact_6563_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_6564_lessThan__def,axiom,
    ( set_or890127255671739683et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_set_nat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6565_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X4: rat] : ( ord_less_rat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6566_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X4: num] : ( ord_less_num @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6567_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X4: int] : ( ord_less_int @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6568_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6569_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X4: real] : ( ord_less_real @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6570_powser__insidea,axiom,
    ! [F: nat > real,X: real,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
       => ( summable_real
          @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_6571_powser__insidea,axiom,
    ! [F: nat > complex,X: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
       => ( summable_real
          @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_6572_lessThan__strict__subset__iff,axiom,
    ! [M2: rat,N: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M2 ) @ ( set_ord_lessThan_rat @ N ) )
      = ( ord_less_rat @ M2 @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6573_lessThan__strict__subset__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M2 ) @ ( set_ord_lessThan_num @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6574_lessThan__strict__subset__iff,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N ) )
      = ( ord_less_int @ M2 @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6575_lessThan__strict__subset__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6576_lessThan__strict__subset__iff,axiom,
    ! [M2: real,N: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M2 ) @ ( set_or5984915006950818249n_real @ N ) )
      = ( ord_less_real @ M2 @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6577_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_6578_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K2: nat] :
      ( ( summable_real @ F )
     => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K2 @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K2 @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K2 ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_6579_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_6580_sum_Onat__diff__reindex,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_6581_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_6582_sum__diff__distrib,axiom,
    ! [Q: real > nat,P: real > nat,N: real] :
      ( ! [X5: real] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
     => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X4: real] : ( minus_minus_nat @ ( P @ X4 ) @ ( Q @ X4 ) )
          @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_6583_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P: nat > nat,N: nat] :
      ( ! [X5: nat] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( minus_minus_nat @ ( P @ X4 ) @ ( Q @ X4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_6584_powser__inside,axiom,
    ! [F: nat > real,X: real,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
       => ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ) ).

% powser_inside
thf(fact_6585_powser__inside,axiom,
    ! [F: nat > complex,X: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
       => ( summable_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ) ).

% powser_inside
thf(fact_6586_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6587_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6588_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6589_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6590_sumr__diff__mult__const2,axiom,
    ! [F: nat > int,N: nat,R2: int] :
      ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_6591_sumr__diff__mult__const2,axiom,
    ! [F: nat > rat,N: nat,R2: rat] :
      ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ R2 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_6592_sumr__diff__mult__const2,axiom,
    ! [F: nat > real,N: nat,R2: real] :
      ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_6593_sum__lessThan__telescope_H,axiom,
    ! [F: nat > rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N3: nat] : ( minus_minus_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6594_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N3: nat] : ( minus_minus_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6595_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6596_sum__lessThan__telescope,axiom,
    ! [F: nat > rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N3: nat] : ( minus_minus_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_rat @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6597_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N3: nat] : ( minus_minus_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_int @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6598_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( minus_minus_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_real @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6599_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_6600_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_6601_power__diff__1__eq,axiom,
    ! [X: complex,N: nat] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex )
      = ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_6602_power__diff__1__eq,axiom,
    ! [X: rat,N: nat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat )
      = ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_6603_power__diff__1__eq,axiom,
    ! [X: int,N: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_6604_power__diff__1__eq,axiom,
    ! [X: real,N: nat] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real )
      = ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_6605_one__diff__power__eq,axiom,
    ! [X: complex,N: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_6606_one__diff__power__eq,axiom,
    ! [X: rat,N: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_6607_one__diff__power__eq,axiom,
    ! [X: int,N: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_6608_one__diff__power__eq,axiom,
    ! [X: real,N: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_6609_geometric__sum,axiom,
    ! [X: complex,N: nat] :
      ( ( X != one_one_complex )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).

% geometric_sum
thf(fact_6610_geometric__sum,axiom,
    ! [X: rat,N: nat] :
      ( ( X != one_one_rat )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).

% geometric_sum
thf(fact_6611_geometric__sum,axiom,
    ! [X: real,N: nat] :
      ( ( X != one_one_real )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).

% geometric_sum
thf(fact_6612_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X4: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X4 @ X4 ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_6613_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X4: real] : ( minus_minus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_6614_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X4: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X4 @ X4 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_6615_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X4: int] : ( minus_minus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_6616_sum__gp__strict,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri8010041392384452111omplex @ N ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_6617_sum__gp__strict,axiom,
    ! [X: rat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri681578069525770553at_rat @ N ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_6618_sum__gp__strict,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri5074537144036343181t_real @ N ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_6619_lemma__termdiff1,axiom,
    ! [Z: complex,H2: complex,M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_complex @ Z @ P5 ) ) @ ( power_power_complex @ Z @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P5 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M2 @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_6620_lemma__termdiff1,axiom,
    ! [Z: rat,H2: rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_rat @ Z @ P5 ) ) @ ( power_power_rat @ Z @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M2 @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_6621_lemma__termdiff1,axiom,
    ! [Z: int,H2: int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_int @ Z @ P5 ) ) @ ( power_power_int @ Z @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M2 @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_6622_lemma__termdiff1,axiom,
    ! [Z: real,H2: real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_real @ Z @ P5 ) ) @ ( power_power_real @ Z @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M2 @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_6623_power__diff__sumr2,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_6624_power__diff__sumr2,axiom,
    ! [X: rat,N: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_rat @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_6625_power__diff__sumr2,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_6626_power__diff__sumr2,axiom,
    ! [X: real,N: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) ) @ ( power_power_real @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_6627_diff__power__eq__sum,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N ) ) @ ( power_power_complex @ Y @ ( suc @ N ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X @ P5 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6628_diff__power__eq__sum,axiom,
    ! [X: rat,N: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N ) ) @ ( power_power_rat @ Y @ ( suc @ N ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X @ P5 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6629_diff__power__eq__sum,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N ) ) @ ( power_power_int @ Y @ ( suc @ N ) ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X @ P5 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6630_diff__power__eq__sum,axiom,
    ! [X: real,N: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N ) ) @ ( power_power_real @ Y @ ( suc @ N ) ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X @ P5 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6631_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > rat,K5: rat,K2: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6632_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > int,K5: int,K2: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6633_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > nat,K5: nat,K2: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6634_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > real,K5: real,K2: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6635_one__diff__power__eq_H,axiom,
    ! [X: complex,N: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6636_one__diff__power__eq_H,axiom,
    ! [X: rat,N: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6637_one__diff__power__eq_H,axiom,
    ! [X: int,N: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6638_one__diff__power__eq_H,axiom,
    ! [X: real,N: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6639_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_split_even_odd
thf(fact_6640_summable__geometric__iff,axiom,
    ! [C: real] :
      ( ( summable_real @ ( power_power_real @ C ) )
      = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_6641_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex @ ( power_power_complex @ C ) )
      = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_6642_summable__divide__iff,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_divide_iff
thf(fact_6643_summable__divide__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_divide_iff
thf(fact_6644_suminf__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( ( suminf_real @ ( power_power_real @ C ) )
        = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% suminf_geometric
thf(fact_6645_suminf__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( ( suminf_complex @ ( power_power_complex @ C ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% suminf_geometric
thf(fact_6646_summable__cmult__iff,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_cmult_iff
thf(fact_6647_summable__cmult__iff,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_cmult_iff
thf(fact_6648_sum__less__suminf2,axiom,
    ! [F: nat > int,N: nat,I: nat] :
      ( ( summable_int @ F )
     => ( ! [M: nat] :
            ( ( ord_less_eq_nat @ N @ M )
           => ( ord_less_eq_int @ zero_zero_int @ ( F @ M ) ) )
       => ( ( ord_less_eq_nat @ N @ I )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_6649_sum__less__suminf2,axiom,
    ! [F: nat > nat,N: nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [M: nat] :
            ( ( ord_less_eq_nat @ N @ M )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M ) ) )
       => ( ( ord_less_eq_nat @ N @ I )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
           => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_6650_sum__less__suminf2,axiom,
    ! [F: nat > real,N: nat,I: nat] :
      ( ( summable_real @ F )
     => ( ! [M: nat] :
            ( ( ord_less_eq_nat @ N @ M )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ M ) ) )
       => ( ( ord_less_eq_nat @ N @ I )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
           => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_6651_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N5 @ N2 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_ratio_test
thf(fact_6652_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N5 @ N2 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_ratio_test
thf(fact_6653_summable__iff__shift,axiom,
    ! [F: nat > real,K2: nat] :
      ( ( summable_real
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K2 ) ) )
      = ( summable_real @ F ) ) ).

% summable_iff_shift
thf(fact_6654_summable__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) ) ) ).

% summable_mult
thf(fact_6655_summable__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C ) ) ) ).

% summable_mult2
thf(fact_6656_summable__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_add
thf(fact_6657_summable__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( summable_nat
          @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_add
thf(fact_6658_summable__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( summable_int
          @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_add
thf(fact_6659_summable__diff,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_diff
thf(fact_6660_summable__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) ) ) ).

% summable_divide
thf(fact_6661_summable__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) ) ) ).

% summable_divide
thf(fact_6662_summable__Suc__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
      = ( summable_real @ F ) ) ).

% summable_Suc_iff
thf(fact_6663_summable__ignore__initial__segment,axiom,
    ! [F: nat > real,K2: nat] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K2 ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_6664_suminf__le,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
     => ( ( summable_real @ F )
       => ( ( summable_real @ G )
         => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).

% suminf_le
thf(fact_6665_suminf__le,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ N2 ) )
     => ( ( summable_nat @ F )
       => ( ( summable_nat @ G )
         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).

% suminf_le
thf(fact_6666_suminf__le,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( G @ N2 ) )
     => ( ( summable_int @ F )
       => ( ( summable_int @ G )
         => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).

% suminf_le
thf(fact_6667_summable__mult__D,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) ) )
     => ( ( C != zero_zero_complex )
       => ( summable_complex @ F ) ) ) ).

% summable_mult_D
thf(fact_6668_summable__mult__D,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
     => ( ( C != zero_zero_real )
       => ( summable_real @ F ) ) ) ).

% summable_mult_D
thf(fact_6669_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_6670_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_6671_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_6672_suminf__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) ) )
        = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).

% suminf_mult
thf(fact_6673_suminf__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( times_times_real @ ( suminf_real @ F ) @ C )
        = ( suminf_real
          @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C ) ) ) ) ).

% suminf_mult2
thf(fact_6674_suminf__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
          = ( suminf_real
            @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_add
thf(fact_6675_suminf__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
          = ( suminf_nat
            @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_add
thf(fact_6676_suminf__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
          = ( suminf_int
            @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_add
thf(fact_6677_suminf__diff,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
          = ( suminf_real
            @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_diff
thf(fact_6678_suminf__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C ) )
        = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_6679_suminf__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C ) )
        = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_6680_suminf__nonneg,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_6681_suminf__nonneg,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_6682_suminf__nonneg,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_6683_suminf__eq__zero__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ( ( suminf_real @ F )
            = zero_zero_real )
          = ( ! [N3: nat] :
                ( ( F @ N3 )
                = zero_zero_real ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_6684_suminf__eq__zero__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ( ( suminf_nat @ F )
            = zero_zero_nat )
          = ( ! [N3: nat] :
                ( ( F @ N3 )
                = zero_zero_nat ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_6685_suminf__eq__zero__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ( ( suminf_int @ F )
            = zero_zero_int )
          = ( ! [N3: nat] :
                ( ( F @ N3 )
                = zero_zero_int ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_6686_suminf__pos,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_pos
thf(fact_6687_suminf__pos,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_pos
thf(fact_6688_suminf__pos,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_pos
thf(fact_6689_summable__zero__power_H,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).

% summable_zero_power'
thf(fact_6690_summable__zero__power_H,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).

% summable_zero_power'
thf(fact_6691_summable__zero__power_H,axiom,
    ! [F: nat > int] :
      ( summable_int
      @ ^ [N3: nat] : ( times_times_int @ ( F @ N3 ) @ ( power_power_int @ zero_zero_int @ N3 ) ) ) ).

% summable_zero_power'
thf(fact_6692_summable__0__powser,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).

% summable_0_powser
thf(fact_6693_summable__0__powser,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).

% summable_0_powser
thf(fact_6694_powser__split__head_I3_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
     => ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).

% powser_split_head(3)
thf(fact_6695_powser__split__head_I3_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
     => ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).

% powser_split_head(3)
thf(fact_6696_summable__powser__split__head,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
      = ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).

% summable_powser_split_head
thf(fact_6697_summable__powser__split__head,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
      = ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).

% summable_powser_split_head
thf(fact_6698_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > complex,M2: nat,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N3 @ M2 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
      = ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_6699_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > real,M2: nat,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N3 @ M2 ) ) @ ( power_power_real @ Z @ N3 ) ) )
      = ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_6700_summable__norm__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N7 @ N2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) ) ).

% summable_norm_comparison_test
thf(fact_6701_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N7 @ N2 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_6702_suminf__pos2,axiom,
    ! [F: nat > real,I: nat] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_6703_suminf__pos2,axiom,
    ! [F: nat > nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_6704_suminf__pos2,axiom,
    ! [F: nat > int,I: nat] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_6705_suminf__pos__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
          = ( ? [I3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_6706_suminf__pos__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
          = ( ? [I3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_6707_suminf__pos__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
          = ( ? [I3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_6708_suminf__le__const,axiom,
    ! [F: nat > int,X: int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X )
       => ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_6709_suminf__le__const,axiom,
    ! [F: nat > nat,X: nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X )
       => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_6710_suminf__le__const,axiom,
    ! [F: nat > real,X: real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X )
       => ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).

% suminf_le_const
thf(fact_6711_summable__comparison__test_H,axiom,
    ! [G: nat > real,N5: nat,F: nat > real] :
      ( ( summable_real @ G )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N5 @ N2 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test'
thf(fact_6712_summable__comparison__test_H,axiom,
    ! [G: nat > real,N5: nat,F: nat > complex] :
      ( ( summable_real @ G )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N5 @ N2 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test'
thf(fact_6713_summable__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N7 @ N2 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test
thf(fact_6714_summable__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N7 @ N2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
     => ( ( summable_real @ G )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test
thf(fact_6715_summableI__nonneg__bounded,axiom,
    ! [F: nat > int,X: int] :
      ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X )
       => ( summable_int @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_6716_summableI__nonneg__bounded,axiom,
    ! [F: nat > nat,X: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X )
       => ( summable_nat @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_6717_summableI__nonneg__bounded,axiom,
    ! [F: nat > real,X: real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X )
       => ( summable_real @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_6718_summable__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ C ) ) ) ).

% summable_geometric
thf(fact_6719_summable__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ C ) ) ) ).

% summable_geometric
thf(fact_6720_complete__algebra__summable__geometric,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ X ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_6721_complete__algebra__summable__geometric,axiom,
    ! [X: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ X ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_6722_suminf__split__head,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_6723_sum__le__suminf,axiom,
    ! [F: nat > int,I6: set_nat] :
      ( ( summable_int @ F )
     => ( ( finite_finite_nat @ I6 )
       => ( ! [N2: nat] :
              ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I6 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I6 ) @ ( suminf_int @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_6724_sum__le__suminf,axiom,
    ! [F: nat > nat,I6: set_nat] :
      ( ( summable_nat @ F )
     => ( ( finite_finite_nat @ I6 )
       => ( ! [N2: nat] :
              ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I6 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) ) )
         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I6 ) @ ( suminf_nat @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_6725_sum__le__suminf,axiom,
    ! [F: nat > real,I6: set_nat] :
      ( ( summable_real @ F )
     => ( ( finite_finite_nat @ I6 )
       => ( ! [N2: nat] :
              ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I6 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) ) )
         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I6 ) @ ( suminf_real @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_6726_suminf__split__initial__segment,axiom,
    ! [F: nat > real,K2: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real @ F )
        = ( plus_plus_real
          @ ( suminf_real
            @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K2 ) ) )
          @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ) ).

% suminf_split_initial_segment
thf(fact_6727_suminf__minus__initial__segment,axiom,
    ! [F: nat > real,K2: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K2 ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ) ).

% suminf_minus_initial_segment
thf(fact_6728_sum__less__suminf,axiom,
    ! [F: nat > int,N: nat] :
      ( ( summable_int @ F )
     => ( ! [M: nat] :
            ( ( ord_less_eq_nat @ N @ M )
           => ( ord_less_int @ zero_zero_int @ ( F @ M ) ) )
       => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_6729_sum__less__suminf,axiom,
    ! [F: nat > nat,N: nat] :
      ( ( summable_nat @ F )
     => ( ! [M: nat] :
            ( ( ord_less_eq_nat @ N @ M )
           => ( ord_less_nat @ zero_zero_nat @ ( F @ M ) ) )
       => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_6730_sum__less__suminf,axiom,
    ! [F: nat > real,N: nat] :
      ( ( summable_real @ F )
     => ( ! [M: nat] :
            ( ( ord_less_eq_nat @ N @ M )
           => ( ord_less_real @ zero_zero_real @ ( F @ M ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_6731_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
     => ( ( suminf_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_6732_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_6733_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) ) )
          @ Z )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_6734_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) ) )
          @ Z )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z @ N3 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_6735_summable__partial__sum__bound,axiom,
    ! [F: nat > complex,E: real] :
      ( ( summable_complex @ F )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ~ ! [N8: nat] :
              ~ ! [M4: nat] :
                  ( ( ord_less_eq_nat @ N8 @ M4 )
                 => ! [N9: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M4 @ N9 ) ) ) @ E ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_6736_summable__partial__sum__bound,axiom,
    ! [F: nat > real,E: real] :
      ( ( summable_real @ F )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ~ ! [N8: nat] :
              ~ ! [M4: nat] :
                  ( ( ord_less_eq_nat @ N8 @ M4 )
                 => ! [N9: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M4 @ N9 ) ) ) @ E ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_6737_suminf__exist__split,axiom,
    ! [R2: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_real @ F )
       => ? [N8: nat] :
          ! [N9: nat] :
            ( ( ord_less_eq_nat @ N8 @ N9 )
           => ( ord_less_real
              @ ( real_V7735802525324610683m_real
                @ ( suminf_real
                  @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N9 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_6738_suminf__exist__split,axiom,
    ! [R2: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_complex @ F )
       => ? [N8: nat] :
          ! [N9: nat] :
            ( ( ord_less_eq_nat @ N8 @ N9 )
           => ( ord_less_real
              @ ( real_V1022390504157884413omplex
                @ ( suminf_complex
                  @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N9 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_6739_summable__power__series,axiom,
    ! [F: nat > real,Z: real] :
      ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( summable_real
              @ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z @ I3 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_6740_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: nat > complex,M7: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
     => ( ( ord_less_real @ R2 @ R0 )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R0 @ N2 ) ) @ M7 )
         => ( summable_real
            @ ^ [N3: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R2 @ N3 ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_6741_sum__bounds__lt__plus1,axiom,
    ! [F: nat > nat,Mm: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_6742_sum__bounds__lt__plus1,axiom,
    ! [F: nat > real,Mm: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_6743_sumr__cos__zero__one,axiom,
    ! [N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ zero_zero_real @ M3 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_6744_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_6745_pred__subset__eq2,axiom,
    ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ( ord_le2162486998276636481er_o_o
        @ ^ [X4: code_integer,Y3: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X4 @ Y3 ) @ R )
        @ ^ [X4: code_integer,Y3: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X4 @ Y3 ) @ S3 ) )
      = ( ord_le8980329558974975238eger_o @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6746_pred__subset__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ( ord_le6124364862034508274_num_o
        @ ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ R )
        @ ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ S3 ) )
      = ( ord_le880128212290418581um_num @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6747_pred__subset__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ( ord_le3404735783095501756_num_o
        @ ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ R )
        @ ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ S3 ) )
      = ( ord_le8085105155179020875at_num @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6748_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
        @ ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S3 ) )
      = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6749_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( ord_le6741204236512500942_int_o
        @ ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ R )
        @ ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ S3 ) )
      = ( ord_le2843351958646193337nt_int @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6750_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6751_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6752_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6753_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6754_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( power_power_real @ Z @ N3 ) )
        @ ( 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_6755_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N3 ) ) @ ( power_power_complex @ Z @ N3 ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_6756_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_6757_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
    = one_one_complex ) ).

% dbl_inc_simps(2)
thf(fact_6758_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_6759_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_6760_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_6761_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_inc_simps(4)
thf(fact_6762_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_inc_simps(4)
thf(fact_6763_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_inc_simps(4)
thf(fact_6764_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_inc_simps(4)
thf(fact_6765_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_6766_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K2 ) )
      = ( numera6690914467698888265omplex @ ( bit1 @ K2 ) ) ) ).

% dbl_inc_simps(5)
thf(fact_6767_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K2 ) )
      = ( numeral_numeral_real @ ( bit1 @ K2 ) ) ) ).

% dbl_inc_simps(5)
thf(fact_6768_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K2 ) )
      = ( numeral_numeral_rat @ ( bit1 @ K2 ) ) ) ).

% dbl_inc_simps(5)
thf(fact_6769_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) ).

% dbl_inc_simps(5)
thf(fact_6770_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) )
      = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K2 ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6771_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6772_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K2 ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K2 ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6773_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) ) )
      = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K2 ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6774_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K2 ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K2 ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6775_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) )
      = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K2 ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6776_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6777_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K2 ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K2 ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6778_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) ) )
      = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K2 ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6779_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K2 ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K2 ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6780_powser__sums__zero__iff,axiom,
    ! [A: nat > complex,X: complex] :
      ( ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_6781_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_6782_sums__le,axiom,
    ! [F: nat > real,G: nat > real,S2: real,T: real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
     => ( ( sums_real @ F @ S2 )
       => ( ( sums_real @ G @ T )
         => ( ord_less_eq_real @ S2 @ T ) ) ) ) ).

% sums_le
thf(fact_6783_sums__le,axiom,
    ! [F: nat > nat,G: nat > nat,S2: nat,T: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ N2 ) )
     => ( ( sums_nat @ F @ S2 )
       => ( ( sums_nat @ G @ T )
         => ( ord_less_eq_nat @ S2 @ T ) ) ) ) ).

% sums_le
thf(fact_6784_sums__le,axiom,
    ! [F: nat > int,G: nat > int,S2: int,T: int] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( G @ N2 ) )
     => ( ( sums_int @ F @ S2 )
       => ( ( sums_int @ G @ T )
         => ( ord_less_eq_int @ S2 @ T ) ) ) ) ).

% sums_le
thf(fact_6785_sums__mult2,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C )
        @ ( times_times_real @ A @ C ) ) ) ).

% sums_mult2
thf(fact_6786_sums__mult,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
        @ ( times_times_real @ C @ A ) ) ) ).

% sums_mult
thf(fact_6787_sums__add,axiom,
    ! [F: nat > real,A: real,G: nat > real,B: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B )
       => ( sums_real
          @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( plus_plus_real @ A @ B ) ) ) ) ).

% sums_add
thf(fact_6788_sums__add,axiom,
    ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
      ( ( sums_nat @ F @ A )
     => ( ( sums_nat @ G @ B )
       => ( sums_nat
          @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% sums_add
thf(fact_6789_sums__add,axiom,
    ! [F: nat > int,A: int,G: nat > int,B: int] :
      ( ( sums_int @ F @ A )
     => ( ( sums_int @ G @ B )
       => ( sums_int
          @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( plus_plus_int @ A @ B ) ) ) ) ).

% sums_add
thf(fact_6790_sums__diff,axiom,
    ! [F: nat > real,A: real,G: nat > real,B: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B )
       => ( sums_real
          @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( minus_minus_real @ A @ B ) ) ) ) ).

% sums_diff
thf(fact_6791_sums__divide,axiom,
    ! [F: nat > complex,A: complex,C: complex] :
      ( ( sums_complex @ F @ A )
     => ( sums_complex
        @ ^ [N3: nat] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ C )
        @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).

% sums_divide
thf(fact_6792_sums__divide,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ C )
        @ ( divide_divide_real @ A @ C ) ) ) ).

% sums_divide
thf(fact_6793_sums__mult2__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ C )
          @ ( times_times_complex @ D @ C ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_6794_sums__mult2__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ C )
          @ ( times_times_real @ D @ C ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_6795_sums__mult__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) )
          @ ( times_times_complex @ C @ D ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_6796_sums__mult__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
          @ ( times_times_real @ C @ D ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_6797_sums__mult__D,axiom,
    ! [C: complex,F: nat > complex,A: complex] :
      ( ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ C @ ( F @ N3 ) )
        @ A )
     => ( ( C != zero_zero_complex )
       => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_6798_sums__mult__D,axiom,
    ! [C: real,F: nat > real,A: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ C @ ( F @ N3 ) )
        @ A )
     => ( ( C != zero_zero_real )
       => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_6799_sums__Suc__imp,axiom,
    ! [F: nat > complex,S2: complex] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
          @ S2 )
       => ( sums_complex @ F @ S2 ) ) ) ).

% sums_Suc_imp
thf(fact_6800_sums__Suc__imp,axiom,
    ! [F: nat > real,S2: real] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( sums_real
          @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
          @ S2 )
       => ( sums_real @ F @ S2 ) ) ) ).

% sums_Suc_imp
thf(fact_6801_sums__Suc__iff,axiom,
    ! [F: nat > real,S2: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ S2 )
      = ( sums_real @ F @ ( plus_plus_real @ S2 @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_6802_sums__Suc,axiom,
    ! [F: nat > real,L: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ L )
     => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6803_sums__Suc,axiom,
    ! [F: nat > nat,L: nat] :
      ( ( sums_nat
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ L )
     => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6804_sums__Suc,axiom,
    ! [F: nat > int,L: int] :
      ( ( sums_int
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ L )
     => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6805_sums__zero__iff__shift,axiom,
    ! [N: nat,F: nat > complex,S2: complex] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ( F @ I2 )
            = zero_zero_complex ) )
     => ( ( sums_complex
          @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
          @ S2 )
        = ( sums_complex @ F @ S2 ) ) ) ).

% sums_zero_iff_shift
thf(fact_6806_sums__zero__iff__shift,axiom,
    ! [N: nat,F: nat > real,S2: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ( F @ I2 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
          @ S2 )
        = ( sums_real @ F @ S2 ) ) ) ).

% sums_zero_iff_shift
thf(fact_6807_powser__sums__if,axiom,
    ! [M2: nat,Z: complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( if_complex @ ( N3 = M2 ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N3 ) )
      @ ( power_power_complex @ Z @ M2 ) ) ).

% powser_sums_if
thf(fact_6808_powser__sums__if,axiom,
    ! [M2: nat,Z: real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( if_real @ ( N3 = M2 ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N3 ) )
      @ ( power_power_real @ Z @ M2 ) ) ).

% powser_sums_if
thf(fact_6809_powser__sums__if,axiom,
    ! [M2: nat,Z: int] :
      ( sums_int
      @ ^ [N3: nat] : ( times_times_int @ ( if_int @ ( N3 = M2 ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N3 ) )
      @ ( power_power_int @ Z @ M2 ) ) ).

% powser_sums_if
thf(fact_6810_powser__sums__zero,axiom,
    ! [A: nat > complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_6811_powser__sums__zero,axiom,
    ! [A: nat > real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_6812_sums__iff__shift,axiom,
    ! [F: nat > real,N: nat,S2: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
        @ S2 )
      = ( sums_real @ F @ ( plus_plus_real @ S2 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% sums_iff_shift
thf(fact_6813_sums__iff__shift_H,axiom,
    ! [F: nat > real,N: nat,S2: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
        @ ( minus_minus_real @ S2 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
      = ( sums_real @ F @ S2 ) ) ).

% sums_iff_shift'
thf(fact_6814_sums__split__initial__segment,axiom,
    ! [F: nat > real,S2: real,N: nat] :
      ( ( sums_real @ F @ S2 )
     => ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N ) )
        @ ( minus_minus_real @ S2 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_6815_sums__If__finite__set_H,axiom,
    ! [G: nat > real,S3: real,A4: set_nat,S4: real,F: nat > real] :
      ( ( sums_real @ G @ S3 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( S4
            = ( plus_plus_real @ S3
              @ ( groups6591440286371151544t_real
                @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
                @ A4 ) ) )
         => ( sums_real
            @ ^ [N3: nat] : ( if_real @ ( member_nat @ N3 @ A4 ) @ ( F @ N3 ) @ ( G @ N3 ) )
            @ S4 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_6816_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_6817_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_6818_pi__half__neq__two,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_neq_two
thf(fact_6819_dbl__inc__def,axiom,
    ( neg_nu8557863876264182079omplex
    = ( ^ [X4: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X4 @ X4 ) @ one_one_complex ) ) ) ).

% dbl_inc_def
thf(fact_6820_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X4: real] : ( plus_plus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_6821_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X4: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X4 @ X4 ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_6822_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X4: int] : ( plus_plus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_6823_pred__equals__eq2,axiom,
    ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ( ( ^ [X4: code_integer,Y3: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X4 @ Y3 ) @ R ) )
        = ( ^ [X4: code_integer,Y3: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X4 @ Y3 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_6824_pred__equals__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ( ( ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ R ) )
        = ( ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_6825_pred__equals__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ( ( ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ R ) )
        = ( ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_6826_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R ) )
        = ( ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_6827_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( ( ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ R ) )
        = ( ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_6828_bot__empty__eq2,axiom,
    ( bot_bo4731626569425807221er_o_o
    = ( ^ [X4: code_integer,Y3: $o] : ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X4 @ Y3 ) @ bot_bo5379713665208646970eger_o ) ) ) ).

% bot_empty_eq2
thf(fact_6829_bot__empty__eq2,axiom,
    ( bot_bot_num_num_o
    = ( ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ bot_bo9056780473022590049um_num ) ) ) ).

% bot_empty_eq2
thf(fact_6830_bot__empty__eq2,axiom,
    ( bot_bot_nat_num_o
    = ( ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ bot_bo7038385379056416535at_num ) ) ) ).

% bot_empty_eq2
thf(fact_6831_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_6832_bot__empty__eq2,axiom,
    ( bot_bot_int_int_o
    = ( ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ bot_bo1796632182523588997nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_6833_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_6834_pi__half__less__two,axiom,
    ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% pi_half_less_two
thf(fact_6835_pi__half__le__two,axiom,
    ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% pi_half_le_two
thf(fact_6836_pi__half__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_gt_zero
thf(fact_6837_pi__half__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_ge_zero
thf(fact_6838_m2pi__less__pi,axiom,
    ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).

% m2pi_less_pi
thf(fact_6839_arctan__ubound,axiom,
    ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_6840_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_6841_geometric__sums,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% geometric_sums
thf(fact_6842_geometric__sums,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% geometric_sums
thf(fact_6843_power__half__series,axiom,
    ( sums_real
    @ ^ [N3: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_6844_subrelI,axiom,
    ! [R2: set_Pr448751882837621926eger_o,S2: set_Pr448751882837621926eger_o] :
      ( ! [X5: code_integer,Y4: $o] :
          ( ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X5 @ Y4 ) @ R2 )
         => ( member1379723562493234055eger_o @ ( produc6677183202524767010eger_o @ X5 @ Y4 ) @ S2 ) )
     => ( ord_le8980329558974975238eger_o @ R2 @ S2 ) ) ).

% subrelI
thf(fact_6845_subrelI,axiom,
    ! [R2: set_Pr8218934625190621173um_num,S2: set_Pr8218934625190621173um_num] :
      ( ! [X5: num,Y4: num] :
          ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ X5 @ Y4 ) @ R2 )
         => ( member7279096912039735102um_num @ ( product_Pair_num_num @ X5 @ Y4 ) @ S2 ) )
     => ( ord_le880128212290418581um_num @ R2 @ S2 ) ) ).

% subrelI
thf(fact_6846_subrelI,axiom,
    ! [R2: set_Pr6200539531224447659at_num,S2: set_Pr6200539531224447659at_num] :
      ( ! [X5: nat,Y4: num] :
          ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X5 @ Y4 ) @ R2 )
         => ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X5 @ Y4 ) @ S2 ) )
     => ( ord_le8085105155179020875at_num @ R2 @ S2 ) ) ).

% subrelI
thf(fact_6847_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X5: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ S2 ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S2 ) ) ).

% subrelI
thf(fact_6848_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X5: int,Y4: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ R2 )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ S2 ) )
     => ( ord_le2843351958646193337nt_int @ R2 @ S2 ) ) ).

% subrelI
thf(fact_6849_minus__pi__half__less__zero,axiom,
    ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).

% minus_pi_half_less_zero
thf(fact_6850_arctan__lbound,axiom,
    ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).

% arctan_lbound
thf(fact_6851_arctan__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arctan_bounded
thf(fact_6852_sums__if_H,axiom,
    ! [G: nat > real,X: real] :
      ( ( sums_real @ G @ X )
     => ( sums_real
        @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X ) ) ).

% sums_if'
thf(fact_6853_sums__if,axiom,
    ! [G: nat > real,X: real,F: nat > real,Y: real] :
      ( ( sums_real @ G @ X )
     => ( ( sums_real @ F @ Y )
       => ( sums_real
          @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( F @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sums_if
thf(fact_6854_machin__Euler,axiom,
    ( ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% machin_Euler
thf(fact_6855_machin,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% machin
thf(fact_6856_sin__cos__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% sin_cos_npi
thf(fact_6857_diffs__equiv,axiom,
    ! [C: nat > complex,X: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X @ N3 ) ) )
     => ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N3 ) @ ( C @ N3 ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X @ N3 ) ) ) ) ) ).

% diffs_equiv
thf(fact_6858_diffs__equiv,axiom,
    ! [C: nat > real,X: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X @ N3 ) ) )
     => ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( C @ N3 ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_real
          @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ).

% diffs_equiv
thf(fact_6859_cos__pi__eq__zero,axiom,
    ! [M2: nat] :
      ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = zero_zero_real ) ).

% cos_pi_eq_zero
thf(fact_6860_monoI1,axiom,
    ! [X9: nat > real] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_real @ ( X9 @ M ) @ ( X9 @ N2 ) ) )
     => ( topolo6980174941875973593q_real @ X9 ) ) ).

% monoI1
thf(fact_6861_monoI1,axiom,
    ! [X9: nat > set_nat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_set_nat @ ( X9 @ M ) @ ( X9 @ N2 ) ) )
     => ( topolo7278393974255667507et_nat @ X9 ) ) ).

% monoI1
thf(fact_6862_monoI1,axiom,
    ! [X9: nat > rat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_rat @ ( X9 @ M ) @ ( X9 @ N2 ) ) )
     => ( topolo4267028734544971653eq_rat @ X9 ) ) ).

% monoI1
thf(fact_6863_monoI1,axiom,
    ! [X9: nat > num] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_num @ ( X9 @ M ) @ ( X9 @ N2 ) ) )
     => ( topolo1459490580787246023eq_num @ X9 ) ) ).

% monoI1
thf(fact_6864_monoI1,axiom,
    ! [X9: nat > nat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_nat @ ( X9 @ M ) @ ( X9 @ N2 ) ) )
     => ( topolo4902158794631467389eq_nat @ X9 ) ) ).

% monoI1
thf(fact_6865_monoI1,axiom,
    ! [X9: nat > int] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_int @ ( X9 @ M ) @ ( X9 @ N2 ) ) )
     => ( topolo4899668324122417113eq_int @ X9 ) ) ).

% monoI1
thf(fact_6866_monoI2,axiom,
    ! [X9: nat > real] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_real @ ( X9 @ N2 ) @ ( X9 @ M ) ) )
     => ( topolo6980174941875973593q_real @ X9 ) ) ).

% monoI2
thf(fact_6867_monoI2,axiom,
    ! [X9: nat > set_nat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_set_nat @ ( X9 @ N2 ) @ ( X9 @ M ) ) )
     => ( topolo7278393974255667507et_nat @ X9 ) ) ).

% monoI2
thf(fact_6868_monoI2,axiom,
    ! [X9: nat > rat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_rat @ ( X9 @ N2 ) @ ( X9 @ M ) ) )
     => ( topolo4267028734544971653eq_rat @ X9 ) ) ).

% monoI2
thf(fact_6869_monoI2,axiom,
    ! [X9: nat > num] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_num @ ( X9 @ N2 ) @ ( X9 @ M ) ) )
     => ( topolo1459490580787246023eq_num @ X9 ) ) ).

% monoI2
thf(fact_6870_monoI2,axiom,
    ! [X9: nat > nat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_nat @ ( X9 @ N2 ) @ ( X9 @ M ) ) )
     => ( topolo4902158794631467389eq_nat @ X9 ) ) ).

% monoI2
thf(fact_6871_monoI2,axiom,
    ! [X9: nat > int] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_int @ ( X9 @ N2 ) @ ( X9 @ M ) ) )
     => ( topolo4899668324122417113eq_int @ X9 ) ) ).

% monoI2
thf(fact_6872_monoseq__def,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X6: nat > real] :
          ( ! [M3: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
             => ( ord_less_eq_real @ ( X6 @ M3 ) @ ( X6 @ N3 ) ) )
          | ! [M3: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
             => ( ord_less_eq_real @ ( X6 @ N3 ) @ ( X6 @ M3 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_6873_monoseq__def,axiom,
    ( topolo7278393974255667507et_nat
    = ( ^ [X6: nat > set_nat] :
          ( ! [M3: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
             => ( ord_less_eq_set_nat @ ( X6 @ M3 ) @ ( X6 @ N3 ) ) )
          | ! [M3: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
             => ( ord_less_eq_set_nat @ ( X6 @ N3 ) @ ( X6 @ M3 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_6874_monoseq__def,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X6: nat > rat] :
          ( ! [M3: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
             => ( ord_less_eq_rat @ ( X6 @ M3 ) @ ( X6 @ N3 ) ) )
          | ! [M3: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
             => ( ord_less_eq_rat @ ( X6 @ N3 ) @ ( X6 @ M3 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_6875_monoseq__def,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X6: nat > num] :
          ( ! [M3: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
             => ( ord_less_eq_num @ ( X6 @ M3 ) @ ( X6 @ N3 ) ) )
          | ! [M3: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
             => ( ord_less_eq_num @ ( X6 @ N3 ) @ ( X6 @ M3 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_6876_monoseq__def,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X6: nat > nat] :
          ( ! [M3: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
             => ( ord_less_eq_nat @ ( X6 @ M3 ) @ ( X6 @ N3 ) ) )
          | ! [M3: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
             => ( ord_less_eq_nat @ ( X6 @ N3 ) @ ( X6 @ M3 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_6877_monoseq__def,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X6: nat > int] :
          ( ! [M3: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
             => ( ord_less_eq_int @ ( X6 @ M3 ) @ ( X6 @ N3 ) ) )
          | ! [M3: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M3 @ N3 )
             => ( ord_less_eq_int @ ( X6 @ N3 ) @ ( X6 @ M3 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_6878_cos__zero,axiom,
    ( ( cos_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cos_zero
thf(fact_6879_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_6880_sin__pi__minus,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
      = ( sin_real @ X ) ) ).

% sin_pi_minus
thf(fact_6881_cos__pi,axiom,
    ( ( cos_real @ pi )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_pi
thf(fact_6882_cos__pi__minus,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_pi_minus
thf(fact_6883_cos__minus__pi,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_minus_pi
thf(fact_6884_sin__minus__pi,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_minus_pi
thf(fact_6885_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_6886_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_6887_sin__npi2,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_6888_sin__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_6889_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_6890_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_6891_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_6892_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_6893_cos__periodic,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cos_real @ X ) ) ).

% cos_periodic
thf(fact_6894_sin__periodic,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( sin_real @ X ) ) ).

% sin_periodic
thf(fact_6895_cos__2pi__minus,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
      = ( cos_real @ X ) ) ).

% cos_2pi_minus
thf(fact_6896_cos__npi,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% cos_npi
thf(fact_6897_cos__npi2,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% cos_npi2
thf(fact_6898_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_6899_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_6900_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_6901_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_6902_sin__2npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
      = zero_zero_real ) ).

% sin_2npi
thf(fact_6903_cos__2npi,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
      = one_one_real ) ).

% cos_2npi
thf(fact_6904_sin__2pi__minus,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_2pi_minus
thf(fact_6905_cos__3over2__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
    = zero_zero_real ) ).

% cos_3over2_pi
thf(fact_6906_sin__3over2__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% sin_3over2_pi
thf(fact_6907_polar__Ex,axiom,
    ! [X: real,Y: real] :
    ? [R4: real,A3: real] :
      ( ( X
        = ( times_times_real @ R4 @ ( cos_real @ A3 ) ) )
      & ( Y
        = ( times_times_real @ R4 @ ( sin_real @ A3 ) ) ) ) ).

% polar_Ex
thf(fact_6908_sin__diff,axiom,
    ! [X: real,Y: real] :
      ( ( sin_real @ ( minus_minus_real @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% sin_diff
thf(fact_6909_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_6910_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_6911_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_6912_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_6913_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_6914_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_6915_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_6916_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
     => ( ( abs_abs_real @ ( cos_real @ X ) )
        = one_one_real ) ) ).

% sin_zero_abs_cos_one
thf(fact_6917_sin__double,axiom,
    ! [X: complex] :
      ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).

% sin_double
thf(fact_6918_sin__double,axiom,
    ! [X: real] :
      ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).

% sin_double
thf(fact_6919_sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).

% sin_le_one
thf(fact_6920_cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).

% cos_le_one
thf(fact_6921_sin__cos__le1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_6922_sin__squared__eq,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_6923_sin__squared__eq,axiom,
    ! [X: real] :
      ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_6924_cos__squared__eq,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_6925_cos__squared__eq,axiom,
    ! [X: real] :
      ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_6926_sin__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).

% sin_ge_minus_one
thf(fact_6927_cos__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).

% cos_ge_minus_one
thf(fact_6928_abs__sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_6929_abs__cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_6930_sin__times__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_6931_sin__times__sin,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_6932_sin__times__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_6933_sin__times__cos,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_6934_cos__times__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_6935_cos__times__sin,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_6936_sin__plus__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_6937_sin__plus__sin,axiom,
    ! [W: real,Z: real] :
      ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_6938_sin__diff__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_6939_sin__diff__sin,axiom,
    ! [W: real,Z: real] :
      ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_6940_cos__diff__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_6941_cos__diff__cos,axiom,
    ! [W: real,Z: real] :
      ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_6942_cos__double,axiom,
    ! [X: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
      = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_6943_cos__double,axiom,
    ! [X: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
      = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_6944_cos__double__sin,axiom,
    ! [W: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_6945_cos__double__sin,axiom,
    ! [W: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_6946_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_6947_diffs__def,axiom,
    ( diffs_int
    = ( ^ [C3: nat > int,N3: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) @ ( C3 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_6948_diffs__def,axiom,
    ( diffs_real
    = ( ^ [C3: nat > real,N3: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( C3 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_6949_diffs__def,axiom,
    ( diffs_rat
    = ( ^ [C3: nat > rat,N3: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) @ ( C3 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_6950_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T5: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T5 )
            & ( ord_less_eq_real @ T5 @ pi )
            & ( X
              = ( cos_real @ T5 ) )
            & ( Y
              = ( sin_real @ T5 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_6951_sin__expansion__lemma,axiom,
    ! [X: real,M2: nat] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_6952_cos__expansion__lemma,axiom,
    ! [X: real,M2: nat] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_6953_termdiff__converges__all,axiom,
    ! [C: nat > complex,X: complex] :
      ( ! [X5: complex] :
          ( summable_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( C @ N3 ) @ ( power_power_complex @ X5 @ N3 ) ) )
     => ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X @ N3 ) ) ) ) ).

% termdiff_converges_all
thf(fact_6954_termdiff__converges__all,axiom,
    ! [C: nat > real,X: real] :
      ( ! [X5: real] :
          ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( C @ N3 ) @ ( power_power_real @ X5 @ N3 ) ) )
     => ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X @ N3 ) ) ) ) ).

% termdiff_converges_all
thf(fact_6955_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero_02
thf(fact_6956_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_6957_cos__is__zero,axiom,
    ? [X5: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X5 )
      & ( ord_less_eq_real @ X5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      & ( ( cos_real @ X5 )
        = zero_zero_real )
      & ! [Y5: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
            & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y5 )
              = zero_zero_real ) )
         => ( Y5 = X5 ) ) ) ).

% cos_is_zero
thf(fact_6958_cos__two__le__zero,axiom,
    ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_le_zero
thf(fact_6959_cos__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X5: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X5 )
            & ( ord_less_eq_real @ X5 @ pi )
            & ( ( cos_real @ X5 )
              = Y )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ pi )
                  & ( ( cos_real @ Y5 )
                    = Y ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% cos_total
thf(fact_6960_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T5: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T5 )
              & ( ord_less_eq_real @ T5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X
                = ( cos_real @ T5 ) )
              & ( Y
                = ( sin_real @ T5 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_6961_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T5 )
          & ( ord_less_eq_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X
            = ( cos_real @ T5 ) )
          & ( Y
            = ( sin_real @ T5 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_6962_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T5: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T5 )
           => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X
                  = ( cos_real @ T5 ) )
               => ( Y
                 != ( sin_real @ T5 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_6963_cos__times__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_6964_cos__times__cos,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_6965_cos__plus__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_6966_cos__plus__cos,axiom,
    ! [W: real,Z: real] :
      ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_6967_sin__gt__zero2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero2
thf(fact_6968_sin__lt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_6969_cos__double__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_6970_sin__30,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_30
thf(fact_6971_cos__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_gt_zero
thf(fact_6972_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X )
                = ( sin_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_6973_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_6974_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_6975_cos__60,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_60
thf(fact_6976_cos__double__cos,axiom,
    ! [W: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).

% cos_double_cos
thf(fact_6977_cos__double__cos,axiom,
    ! [W: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).

% cos_double_cos
thf(fact_6978_cos__treble__cos,axiom,
    ! [X: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).

% cos_treble_cos
thf(fact_6979_cos__treble__cos,axiom,
    ! [X: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).

% cos_treble_cos
thf(fact_6980_termdiff__converges,axiom,
    ! [X: real,K5: real,C: nat > real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ K5 )
     => ( ! [X5: real] :
            ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X5 ) @ K5 )
           => ( summable_real
              @ ^ [N3: nat] : ( times_times_real @ ( C @ N3 ) @ ( power_power_real @ X5 @ N3 ) ) ) )
       => ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ).

% termdiff_converges
thf(fact_6981_termdiff__converges,axiom,
    ! [X: complex,K5: real,C: nat > complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ K5 )
     => ( ! [X5: complex] :
            ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X5 ) @ K5 )
           => ( summable_complex
              @ ^ [N3: nat] : ( times_times_complex @ ( C @ N3 ) @ ( power_power_complex @ X5 @ N3 ) ) ) )
       => ( summable_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X @ N3 ) ) ) ) ) ).

% termdiff_converges
thf(fact_6982_sin__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_6983_sin__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_6984_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_6985_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_6986_sin__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X5: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
            & ( ord_less_eq_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X5 )
              = Y )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y5 )
                    = Y ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% sin_total
thf(fact_6987_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_6988_cos__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_ge_zero
thf(fact_6989_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X4: nat] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X4: nat] :
            ( X
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_6990_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_6991_sin__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
       => ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_6992_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_6993_cos__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( cos_real @ X )
          = zero_zero_real )
       => ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_6994_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_6995_mono__SucI1,axiom,
    ! [X9: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( X9 @ N2 ) @ ( X9 @ ( suc @ N2 ) ) )
     => ( topolo6980174941875973593q_real @ X9 ) ) ).

% mono_SucI1
thf(fact_6996_mono__SucI1,axiom,
    ! [X9: nat > set_nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( X9 @ N2 ) @ ( X9 @ ( suc @ N2 ) ) )
     => ( topolo7278393974255667507et_nat @ X9 ) ) ).

% mono_SucI1
thf(fact_6997_mono__SucI1,axiom,
    ! [X9: nat > rat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( X9 @ N2 ) @ ( X9 @ ( suc @ N2 ) ) )
     => ( topolo4267028734544971653eq_rat @ X9 ) ) ).

% mono_SucI1
thf(fact_6998_mono__SucI1,axiom,
    ! [X9: nat > num] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( X9 @ N2 ) @ ( X9 @ ( suc @ N2 ) ) )
     => ( topolo1459490580787246023eq_num @ X9 ) ) ).

% mono_SucI1
thf(fact_6999_mono__SucI1,axiom,
    ! [X9: nat > nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( X9 @ N2 ) @ ( X9 @ ( suc @ N2 ) ) )
     => ( topolo4902158794631467389eq_nat @ X9 ) ) ).

% mono_SucI1
thf(fact_7000_mono__SucI1,axiom,
    ! [X9: nat > int] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( X9 @ N2 ) @ ( X9 @ ( suc @ N2 ) ) )
     => ( topolo4899668324122417113eq_int @ X9 ) ) ).

% mono_SucI1
thf(fact_7001_mono__SucI2,axiom,
    ! [X9: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( X9 @ ( suc @ N2 ) ) @ ( X9 @ N2 ) )
     => ( topolo6980174941875973593q_real @ X9 ) ) ).

% mono_SucI2
thf(fact_7002_mono__SucI2,axiom,
    ! [X9: nat > set_nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( X9 @ ( suc @ N2 ) ) @ ( X9 @ N2 ) )
     => ( topolo7278393974255667507et_nat @ X9 ) ) ).

% mono_SucI2
thf(fact_7003_mono__SucI2,axiom,
    ! [X9: nat > rat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( X9 @ ( suc @ N2 ) ) @ ( X9 @ N2 ) )
     => ( topolo4267028734544971653eq_rat @ X9 ) ) ).

% mono_SucI2
thf(fact_7004_mono__SucI2,axiom,
    ! [X9: nat > num] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( X9 @ ( suc @ N2 ) ) @ ( X9 @ N2 ) )
     => ( topolo1459490580787246023eq_num @ X9 ) ) ).

% mono_SucI2
thf(fact_7005_mono__SucI2,axiom,
    ! [X9: nat > nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( X9 @ ( suc @ N2 ) ) @ ( X9 @ N2 ) )
     => ( topolo4902158794631467389eq_nat @ X9 ) ) ).

% mono_SucI2
thf(fact_7006_mono__SucI2,axiom,
    ! [X9: nat > int] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( X9 @ ( suc @ N2 ) ) @ ( X9 @ N2 ) )
     => ( topolo4899668324122417113eq_int @ X9 ) ) ).

% mono_SucI2
thf(fact_7007_monoseq__Suc,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X6: nat > real] :
          ( ! [N3: nat] : ( ord_less_eq_real @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_real @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7008_monoseq__Suc,axiom,
    ( topolo7278393974255667507et_nat
    = ( ^ [X6: nat > set_nat] :
          ( ! [N3: nat] : ( ord_less_eq_set_nat @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_set_nat @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7009_monoseq__Suc,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X6: nat > rat] :
          ( ! [N3: nat] : ( ord_less_eq_rat @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_rat @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7010_monoseq__Suc,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X6: nat > num] :
          ( ! [N3: nat] : ( ord_less_eq_num @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_num @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7011_monoseq__Suc,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X6: nat > nat] :
          ( ! [N3: nat] : ( ord_less_eq_nat @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_nat @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7012_monoseq__Suc,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X6: nat > int] :
          ( ! [N3: nat] : ( ord_less_eq_int @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_int @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7013_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ? [T5: real] :
            ( ( ord_less_real @ X @ T5 )
            & ( ord_less_real @ T5 @ zero_zero_real )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_7014_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T5: real] :
            ( ( ord_less_real @ zero_zero_real @ T5 )
            & ( ord_less_real @ T5 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_7015_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T5: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
      & ( ( cos_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_7016_sin__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
      @ ( sin_real @ X ) ) ).

% sin_paired
thf(fact_7017_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_7018_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_7019_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T5: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T5 )
           => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T5 ) @ ( sin_real @ T5 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_7020_fact__0,axiom,
    ( ( semiri5044797733671781792omplex @ zero_zero_nat )
    = one_one_complex ) ).

% fact_0
thf(fact_7021_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_7022_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_7023_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_7024_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_7025_fact__1,axiom,
    ( ( semiri5044797733671781792omplex @ one_one_nat )
    = one_one_complex ) ).

% fact_1
thf(fact_7026_fact__1,axiom,
    ( ( semiri773545260158071498ct_rat @ one_one_nat )
    = one_one_rat ) ).

% fact_1
thf(fact_7027_fact__1,axiom,
    ( ( semiri1406184849735516958ct_int @ one_one_nat )
    = one_one_int ) ).

% fact_1
thf(fact_7028_fact__1,axiom,
    ( ( semiri2265585572941072030t_real @ one_one_nat )
    = one_one_real ) ).

% fact_1
thf(fact_7029_fact__1,axiom,
    ( ( semiri1408675320244567234ct_nat @ one_one_nat )
    = one_one_nat ) ).

% fact_1
thf(fact_7030_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_7031_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_7032_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_7033_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_7034_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_7035_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_Suc
thf(fact_7036_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_Suc
thf(fact_7037_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_Suc
thf(fact_7038_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_Suc
thf(fact_7039_tan__npi,axiom,
    ! [N: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_7040_tan__periodic__n,axiom,
    ! [X: real,N: num] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_n
thf(fact_7041_tan__periodic__nat,axiom,
    ! [X: real,N: nat] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_nat
thf(fact_7042_fact__2,axiom,
    ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7043_fact__2,axiom,
    ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7044_fact__2,axiom,
    ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7045_fact__2,axiom,
    ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7046_fact__2,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7047_norm__cos__sin,axiom,
    ! [T: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
      = one_one_real ) ).

% norm_cos_sin
thf(fact_7048_tan__periodic,axiom,
    ! [X: real] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic
thf(fact_7049_complex__diff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).

% complex_diff
thf(fact_7050_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_zero
thf(fact_7051_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_zero
thf(fact_7052_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_zero
thf(fact_7053_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_zero
thf(fact_7054_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_gt_zero
thf(fact_7055_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_gt_zero
thf(fact_7056_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_gt_zero
thf(fact_7057_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_gt_zero
thf(fact_7058_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_7059_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_7060_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_7061_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_7062_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_1
thf(fact_7063_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_1
thf(fact_7064_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_1
thf(fact_7065_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_1
thf(fact_7066_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_mono
thf(fact_7067_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M2 ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_mono
thf(fact_7068_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M2 ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_mono
thf(fact_7069_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono
thf(fact_7070_fact__dvd,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M2 ) ) ) ).

% fact_dvd
thf(fact_7071_fact__dvd,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M2 ) ) ) ).

% fact_dvd
thf(fact_7072_fact__dvd,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M2 ) ) ) ).

% fact_dvd
thf(fact_7073_fact__dvd,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M2 ) ) ) ).

% fact_dvd
thf(fact_7074_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).

% fact_less_mono
thf(fact_7075_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M2 ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).

% fact_less_mono
thf(fact_7076_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_real @ ( semiri2265585572941072030t_real @ M2 ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).

% fact_less_mono
thf(fact_7077_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono
thf(fact_7078_fact__mod,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M2 ) )
        = zero_zero_int ) ) ).

% fact_mod
thf(fact_7079_fact__mod,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M2 ) )
        = zero_zero_nat ) ) ).

% fact_mod
thf(fact_7080_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K2 ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K2 @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7081_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K2 @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7082_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K2 ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K2 @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7083_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K2 @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7084_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K2 @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7085_fact__le__power,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_7086_fact__le__power,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_7087_fact__le__power,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_7088_fact__le__power,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_7089_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).

% complex_mult
thf(fact_7090_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = one_one_complex )
      = ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
thf(fact_7091_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_7092_tan__def,axiom,
    ( tan_complex
    = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X4 ) @ ( cos_complex @ X4 ) ) ) ) ).

% tan_def
thf(fact_7093_tan__def,axiom,
    ( tan_real
    = ( ^ [X4: real] : ( divide_divide_real @ ( sin_real @ X4 ) @ ( cos_real @ X4 ) ) ) ) ).

% tan_def
thf(fact_7094_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K2 ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).

% choose_dvd
thf(fact_7095_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% choose_dvd
thf(fact_7096_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% choose_dvd
thf(fact_7097_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% choose_dvd
thf(fact_7098_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% choose_dvd
thf(fact_7099_fact__numeral,axiom,
    ! [K2: num] :
      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K2 ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ K2 ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K2 ) ) ) ) ).

% fact_numeral
thf(fact_7100_fact__numeral,axiom,
    ! [K2: num] :
      ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K2 ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K2 ) ) ) ) ).

% fact_numeral
thf(fact_7101_fact__numeral,axiom,
    ! [K2: num] :
      ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K2 ) )
      = ( times_times_int @ ( numeral_numeral_int @ K2 ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K2 ) ) ) ) ).

% fact_numeral
thf(fact_7102_fact__numeral,axiom,
    ! [K2: num] :
      ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K2 ) )
      = ( times_times_real @ ( numeral_numeral_real @ K2 ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K2 ) ) ) ) ).

% fact_numeral
thf(fact_7103_fact__numeral,axiom,
    ! [K2: num] :
      ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K2 ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K2 ) ) ) ) ).

% fact_numeral
thf(fact_7104_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( A
          = ( uminus_uminus_real @ one_one_real ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
thf(fact_7105_square__fact__le__2__fact,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% square_fact_le_2_fact
thf(fact_7106_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_7107_fact__num__eq__if,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M3 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7108_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7109_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M3 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7110_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7111_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7112_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri1406184849735516958ct_int @ N )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7113_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri773545260158071498ct_rat @ N )
        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7114_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri2265585572941072030t_real @ N )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7115_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri1408675320244567234ct_nat @ N )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7116_tan__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).

% tan_gt_zero
thf(fact_7117_lemma__tan__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X5: real] :
          ( ( ord_less_real @ zero_zero_real @ X5 )
          & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y @ ( tan_real @ X5 ) ) ) ) ).

% lemma_tan_total
thf(fact_7118_tan__total,axiom,
    ! [Y: real] :
    ? [X5: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
      & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X5 )
        = Y )
      & ! [Y5: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
            & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y5 )
              = Y ) )
         => ( Y5 = X5 ) ) ) ).

% tan_total
thf(fact_7119_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).

% tan_monotone
thf(fact_7120_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
         => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ Y @ X )
              = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_7121_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
              = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_7122_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X5: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
      & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X5 )
        = Y ) ) ).

% lemma_tan_total1
thf(fact_7123_tan__minus__45,axiom,
    ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% tan_minus_45
thf(fact_7124_tan__inverse,axiom,
    ! [Y: real] :
      ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y ) )
      = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y ) ) ) ).

% tan_inverse
thf(fact_7125_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_7126_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_7127_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_7128_tan__total__pos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ? [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
          & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X5 )
            = Y ) ) ) ).

% tan_total_pos
thf(fact_7129_tan__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_7130_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_7131_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).

% tan_mono_le
thf(fact_7132_tan__bound__pi2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
     => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).

% tan_bound_pi2
thf(fact_7133_arctan,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ ( arctan @ Y ) )
        = Y ) ) ).

% arctan
thf(fact_7134_arctan__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arctan @ ( tan_real @ X ) )
          = X ) ) ) ).

% arctan_tan
thf(fact_7135_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ( tan_real @ X )
            = Y )
         => ( ( arctan @ Y )
            = X ) ) ) ) ).

% arctan_unique
thf(fact_7136_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > complex > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).

% Maclaurin_zero
thf(fact_7137_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > real > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).

% Maclaurin_zero
thf(fact_7138_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > rat > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).

% Maclaurin_zero
thf(fact_7139_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > nat > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).

% Maclaurin_zero
thf(fact_7140_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > int > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).

% Maclaurin_zero
thf(fact_7141_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B8: real] :
          ( ( F @ H2 )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
              @ ( set_ord_lessThan_nat @ N ) )
            @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_7142_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_7143_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_7144_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_7145_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_7146_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_7147_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_7148_tan__total__pi4,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ? [Z3: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z3 )
          & ( ord_less_real @ Z3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z3 )
            = X ) ) ) ).

% tan_total_pi4
thf(fact_7149_tan__half,axiom,
    ( tan_complex
    = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_complex ) ) ) ) ).

% tan_half
thf(fact_7150_tan__half,axiom,
    ( tan_real
    = ( ^ [X4: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_real ) ) ) ) ).

% tan_half
thf(fact_7151_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_7152_cos__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      @ ( cos_real @ X ) ) ).

% cos_paired
thf(fact_7153_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ? [T5: real] :
            ( ( ord_less_real @ zero_zero_real @ T5 )
            & ( ord_less_real @ T5 @ X )
            & ( ( sin_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_7154_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [T5: real] :
          ( ( ord_less_real @ zero_zero_real @ T5 )
          & ( ord_less_eq_real @ T5 @ X )
          & ( ( sin_real @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_7155_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T5: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
      & ( ( sin_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_7156_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T5: real] :
      ( ( sin_real @ X )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_7157_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_7158_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T5: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
            & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
            & ( ( exp_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_7159_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_7160_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_7161_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ X )
        = one_one_real )
      = ( X = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_7162_one__less__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% one_less_exp_iff
thf(fact_7163_exp__less__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_7164_exp__le__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_7165_one__le__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% one_le_exp_iff
thf(fact_7166_fact__ge__self,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_self
thf(fact_7167_fact__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono_nat
thf(fact_7168_exp__times__arg__commute,axiom,
    ! [A4: complex] :
      ( ( times_times_complex @ ( exp_complex @ A4 ) @ A4 )
      = ( times_times_complex @ A4 @ ( exp_complex @ A4 ) ) ) ).

% exp_times_arg_commute
thf(fact_7169_exp__times__arg__commute,axiom,
    ! [A4: real] :
      ( ( times_times_real @ ( exp_real @ A4 ) @ A4 )
      = ( times_times_real @ A4 @ ( exp_real @ A4 ) ) ) ).

% exp_times_arg_commute
thf(fact_7170_fact__less__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono_nat
thf(fact_7171_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_7172_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_7173_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_7174_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_7175_exp__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( exp_complex @ ( minus_minus_complex @ X @ Y ) )
      = ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ).

% exp_diff
thf(fact_7176_exp__diff,axiom,
    ! [X: real,Y: real] :
      ( ( exp_real @ ( minus_minus_real @ X @ Y ) )
      = ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).

% exp_diff
thf(fact_7177_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_Suc_0_nat
thf(fact_7178_exp__gt__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).

% exp_gt_one
thf(fact_7179_exp__ge__add__one__self,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).

% exp_ge_add_one_self
thf(fact_7180_dvd__fact,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_nat @ M2 @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% dvd_fact
thf(fact_7181_exp__minus__inverse,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
      = one_one_real ) ).

% exp_minus_inverse
thf(fact_7182_exp__minus__inverse,axiom,
    ! [X: complex] :
      ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
      = one_one_complex ) ).

% exp_minus_inverse
thf(fact_7183_exp__of__nat2__mult,axiom,
    ! [X: complex,N: nat] :
      ( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N ) ) )
      = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).

% exp_of_nat2_mult
thf(fact_7184_exp__of__nat2__mult,axiom,
    ! [X: real,N: nat] :
      ( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N ) ) )
      = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).

% exp_of_nat2_mult
thf(fact_7185_exp__of__nat__mult,axiom,
    ! [N: nat,X: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X ) )
      = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).

% exp_of_nat_mult
thf(fact_7186_exp__of__nat__mult,axiom,
    ! [N: nat,X: real] :
      ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) )
      = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).

% exp_of_nat_mult
thf(fact_7187_fact__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ ( suc @ M2 ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ N ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_7188_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_7189_lemma__exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y )
     => ? [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
          & ( ord_less_eq_real @ X5 @ ( minus_minus_real @ Y @ one_one_real ) )
          & ( ( exp_real @ X5 )
            = Y ) ) ) ).

% lemma_exp_total
thf(fact_7190_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( ( ord_less_eq_nat @ R2 @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_7191_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_7192_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_7193_exp__divide__power__eq,axiom,
    ! [N: nat,X: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
        = ( exp_complex @ X ) ) ) ).

% exp_divide_power_eq
thf(fact_7194_exp__divide__power__eq,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
        = ( exp_real @ X ) ) ) ).

% exp_divide_power_eq
thf(fact_7195_tanh__altdef,axiom,
    ( tanh_real
    = ( ^ [X4: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) @ ( plus_plus_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_7196_tanh__altdef,axiom,
    ( tanh_complex
    = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_7197_exp__half__le2,axiom,
    ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% exp_half_le2
thf(fact_7198_exp__double,axiom,
    ! [Z: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
      = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_7199_exp__double,axiom,
    ! [Z: real] :
      ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
      = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_7200_sin__coeff__Suc,axiom,
    ! [N: nat] :
      ( ( sin_coeff @ ( suc @ N ) )
      = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).

% sin_coeff_Suc
thf(fact_7201_exp__bound__half,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_7202_exp__bound__half,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_7203_exp__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_7204_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_7205_cos__coeff__Suc,axiom,
    ! [N: nat] :
      ( ( cos_coeff @ ( suc @ N ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).

% cos_coeff_Suc
thf(fact_7206_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_7207_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_7208_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_7209_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_7210_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T5: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
      & ( ( exp_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_7211_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_7212_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X4: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_7213_sin__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( sin_real @ X )
        = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_7214_cos__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( cos_real @ X )
        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_7215_pochhammer__double,axiom,
    ! [Z: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_7216_pochhammer__double,axiom,
    ! [Z: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_7217_pochhammer__double,axiom,
    ! [Z: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_7218_of__nat__code,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N3: nat] :
          ( semiri2816024913162550771omplex
          @ ^ [I3: complex] : ( plus_plus_complex @ I3 @ one_one_complex )
          @ N3
          @ zero_zero_complex ) ) ) ).

% of_nat_code
thf(fact_7219_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N3: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
          @ N3
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_7220_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N3: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I3: real] : ( plus_plus_real @ I3 @ one_one_real )
          @ N3
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_7221_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N3: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
          @ N3
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_7222_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N3: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I3: rat] : ( plus_plus_rat @ I3 @ one_one_rat )
          @ N3
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_7223_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M2 ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M2 ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M2 ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_7224_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M2 ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M2 ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M2 ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_7225_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M2 ) )
      = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M2 ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_7226_real__sqrt__one,axiom,
    ( ( sqrt @ one_one_real )
    = one_one_real ) ).

% real_sqrt_one
thf(fact_7227_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( sqrt @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% real_sqrt_eq_1_iff
thf(fact_7228_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K2 ) )
      = zero_zero_complex ) ).

% gbinomial_0(2)
thf(fact_7229_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_real @ zero_zero_real @ ( suc @ K2 ) )
      = zero_zero_real ) ).

% gbinomial_0(2)
thf(fact_7230_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K2 ) )
      = zero_zero_rat ) ).

% gbinomial_0(2)
thf(fact_7231_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K2 ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_7232_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K2 ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_7233_gbinomial__0_I1_J,axiom,
    ! [A: complex] :
      ( ( gbinomial_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% gbinomial_0(1)
thf(fact_7234_gbinomial__0_I1_J,axiom,
    ! [A: real] :
      ( ( gbinomial_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% gbinomial_0(1)
thf(fact_7235_gbinomial__0_I1_J,axiom,
    ! [A: rat] :
      ( ( gbinomial_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% gbinomial_0(1)
thf(fact_7236_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_7237_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_7238_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% real_sqrt_lt_1_iff
thf(fact_7239_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_real @ one_one_real @ Y ) ) ).

% real_sqrt_gt_1_iff
thf(fact_7240_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_7241_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ one_one_real @ Y ) ) ).

% real_sqrt_ge_1_iff
thf(fact_7242_pochhammer__0,axiom,
    ! [A: complex] :
      ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% pochhammer_0
thf(fact_7243_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_7244_pochhammer__0,axiom,
    ! [A: rat] :
      ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_7245_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_7246_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_7247_real__sqrt__abs2,axiom,
    ! [X: real] :
      ( ( sqrt @ ( times_times_real @ X @ X ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs2
thf(fact_7248_real__sqrt__mult__self,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
      = ( abs_abs_real @ A ) ) ).

% real_sqrt_mult_self
thf(fact_7249_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_7250_real__sqrt__abs,axiom,
    ! [X: real] :
      ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs
thf(fact_7251_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X ) ) ).

% real_sqrt_pow2
thf(fact_7252_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% real_sqrt_pow2_iff
thf(fact_7253_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] :
      ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_7254_real__sqrt__mult,axiom,
    ! [X: real,Y: real] :
      ( ( sqrt @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_mult
thf(fact_7255_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_one
thf(fact_7256_pochhammer__pos,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_7257_pochhammer__pos,axiom,
    ! [X: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_7258_pochhammer__pos,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_7259_pochhammer__pos,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_7260_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M2: nat,N: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ M2 )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s2602460028002588243omplex @ A @ N )
         != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_7261_pochhammer__neq__0__mono,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ M2 )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s7457072308508201937r_real @ A @ N )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_7262_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ M2 )
       != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s4028243227959126397er_rat @ A @ N )
         != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_7263_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N: nat,M2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N )
        = zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s2602460028002588243omplex @ A @ M2 )
          = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_7264_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s7457072308508201937r_real @ A @ M2 )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_7265_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N )
        = zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s4028243227959126397er_rat @ A @ M2 )
          = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_7266_pochhammer__fact,axiom,
    ( semiri5044797733671781792omplex
    = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).

% pochhammer_fact
thf(fact_7267_pochhammer__fact,axiom,
    ( semiri773545260158071498ct_rat
    = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).

% pochhammer_fact
thf(fact_7268_pochhammer__fact,axiom,
    ( semiri1406184849735516958ct_int
    = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).

% pochhammer_fact
thf(fact_7269_pochhammer__fact,axiom,
    ( semiri2265585572941072030t_real
    = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).

% pochhammer_fact
thf(fact_7270_pochhammer__fact,axiom,
    ( semiri1408675320244567234ct_nat
    = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).

% pochhammer_fact
thf(fact_7271_gbinomial__pochhammer,axiom,
    ( gbinomial_complex
    = ( ^ [A5: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A5 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_7272_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A5 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_7273_gbinomial__pochhammer,axiom,
    ( gbinomial_real
    = ( ^ [A5: real,K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A5 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_7274_gbinomial__pochhammer_H,axiom,
    ( gbinomial_complex
    = ( ^ [A5: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A5 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_7275_gbinomial__pochhammer_H,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_7276_gbinomial__pochhammer_H,axiom,
    ( gbinomial_real
    = ( ^ [A5: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A5 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_7277_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_7278_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K2: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ A @ K2 ) @ ( gbinomial_complex @ A @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_7279_gbinomial__Suc__Suc,axiom,
    ! [A: real,K2: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K2 ) )
      = ( plus_plus_real @ ( gbinomial_real @ A @ K2 ) @ ( gbinomial_real @ A @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_7280_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ A @ K2 ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_7281_gbinomial__of__nat__symmetric,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K2 )
        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_7282_gbinomial__of__nat__symmetric,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K2 )
        = ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_7283_pochhammer__nonneg,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_7284_pochhammer__nonneg,axiom,
    ! [X: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_7285_pochhammer__nonneg,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_7286_pochhammer__nonneg,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_7287_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
          = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
          = zero_zero_complex ) ) ) ).

% pochhammer_0_left
thf(fact_7288_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% pochhammer_0_left
thf(fact_7289_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% pochhammer_0_left
thf(fact_7290_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% pochhammer_0_left
thf(fact_7291_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% pochhammer_0_left
thf(fact_7292_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_7293_gbinomial__addition__formula,axiom,
    ! [A: complex,K2: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K2 ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K2 ) ) ) ).

% gbinomial_addition_formula
thf(fact_7294_gbinomial__addition__formula,axiom,
    ! [A: real,K2: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K2 ) )
      = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K2 ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K2 ) ) ) ).

% gbinomial_addition_formula
thf(fact_7295_gbinomial__addition__formula,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K2 ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K2 ) ) ) ).

% gbinomial_addition_formula
thf(fact_7296_gbinomial__absorb__comp,axiom,
    ! [A: complex,K2: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ ( gbinomial_complex @ A @ K2 ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K2 ) ) ) ).

% gbinomial_absorb_comp
thf(fact_7297_gbinomial__absorb__comp,axiom,
    ! [A: real,K2: nat] :
      ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( gbinomial_real @ A @ K2 ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K2 ) ) ) ).

% gbinomial_absorb_comp
thf(fact_7298_gbinomial__absorb__comp,axiom,
    ! [A: rat,K2: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( gbinomial_rat @ A @ K2 ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K2 ) ) ) ).

% gbinomial_absorb_comp
thf(fact_7299_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K2 ) @ A )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 ) @ ( gbinomial_real @ A @ K2 ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_7300_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: rat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 ) @ ( gbinomial_rat @ A @ K2 ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_7301_gbinomial__mult__1,axiom,
    ! [A: real,K2: nat] :
      ( ( times_times_real @ A @ ( gbinomial_real @ A @ K2 ) )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( gbinomial_real @ A @ K2 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) @ ( gbinomial_real @ A @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_7302_gbinomial__mult__1,axiom,
    ! [A: rat,K2: nat] :
      ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( gbinomial_rat @ A @ K2 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_7303_gbinomial__mult__1_H,axiom,
    ! [A: real,K2: nat] :
      ( ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ A )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( gbinomial_real @ A @ K2 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) @ ( gbinomial_real @ A @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_7304_gbinomial__mult__1_H,axiom,
    ! [A: rat,K2: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ A )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( gbinomial_rat @ A @ K2 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_7305_pochhammer__rec,axiom,
    ! [A: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
      = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_7306_pochhammer__rec,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_7307_pochhammer__rec,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_7308_pochhammer__rec,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_7309_pochhammer__rec,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_7310_pochhammer__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_7311_pochhammer__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_7312_pochhammer__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_7313_pochhammer__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_7314_pochhammer__rec_H,axiom,
    ! [Z: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
      = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_7315_pochhammer__rec_H,axiom,
    ! [Z: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
      = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_7316_pochhammer__rec_H,axiom,
    ! [Z: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_7317_pochhammer__rec_H,axiom,
    ! [Z: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_7318_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N )
        = zero_zero_complex )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7319_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N )
        = zero_zero_real )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7320_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N )
        = zero_zero_rat )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7321_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K2 )
        = zero_zero_complex )
      = ( ord_less_nat @ N @ K2 ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7322_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K2 )
        = zero_z3403309356797280102nteger )
      = ( ord_less_nat @ N @ K2 ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7323_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K2 )
        = zero_zero_int )
      = ( ord_less_nat @ N @ K2 ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7324_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K2 )
        = zero_zero_real )
      = ( ord_less_nat @ N @ K2 ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7325_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K2 )
        = zero_zero_rat )
      = ( ord_less_nat @ N @ K2 ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7326_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ N @ K2 )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K2 )
        = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7327_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ N @ K2 )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K2 )
        = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7328_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ N @ K2 )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K2 )
        = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7329_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ N @ K2 )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K2 )
        = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7330_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ N @ K2 )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K2 )
        = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7331_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K2 )
       != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7332_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K2 )
       != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7333_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K2 )
       != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7334_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K2 )
       != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7335_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K2 )
       != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7336_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_7337_pochhammer__product_H,axiom,
    ! [Z: real,N: nat,M2: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_7338_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_7339_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M2: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_7340_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).

% real_less_rsqrt
thf(fact_7341_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
     => ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_7342_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).

% real_le_rsqrt
thf(fact_7343_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K2 ) ) ) ).

% Suc_times_gbinomial
thf(fact_7344_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K2 ) ) )
      = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K2 ) ) ) ).

% Suc_times_gbinomial
thf(fact_7345_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K2 ) ) ) ).

% Suc_times_gbinomial
thf(fact_7346_gbinomial__absorption,axiom,
    ! [K2: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) @ ( gbinomial_complex @ A @ ( suc @ K2 ) ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K2 ) ) ) ).

% gbinomial_absorption
thf(fact_7347_gbinomial__absorption,axiom,
    ! [K2: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) @ ( gbinomial_real @ A @ ( suc @ K2 ) ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K2 ) ) ) ).

% gbinomial_absorption
thf(fact_7348_gbinomial__absorption,axiom,
    ! [K2: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K2 ) ) ) ).

% gbinomial_absorption
thf(fact_7349_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,M2: nat,A: real] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( times_times_real @ ( gbinomial_real @ A @ M2 ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M2 ) @ K2 ) )
        = ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( minus_minus_nat @ M2 @ K2 ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_7350_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,M2: nat,A: rat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( times_times_rat @ ( gbinomial_rat @ A @ M2 ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M2 ) @ K2 ) )
        = ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( minus_minus_nat @ M2 @ K2 ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_7351_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4660882817536571857er_int @ Z @ N )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_7352_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s7457072308508201937r_real @ Z @ N )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_7353_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4663373288045622133er_nat @ Z @ N )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_7354_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z: rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4028243227959126397er_rat @ Z @ N )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_7355_real__le__lsqrt,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 @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_le_lsqrt
thf(fact_7356_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( sqrt @ X )
          = Y ) ) ) ).

% real_sqrt_unique
thf(fact_7357_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( ( ord_less_real @ zero_zero_real @ U )
     => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).

% lemma_real_divide_sqrt_less
thf(fact_7358_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = X )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_7359_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = Y )
     => ( X = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_7360_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_7361_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_7362_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_7363_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
     => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).

% sqrt_ge_absD
thf(fact_7364_cos__45,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_45
thf(fact_7365_sin__45,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_45
thf(fact_7366_gbinomial__factors,axiom,
    ! [A: complex,K2: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) ) @ ( gbinomial_complex @ A @ K2 ) ) ) ).

% gbinomial_factors
thf(fact_7367_gbinomial__factors,axiom,
    ! [A: real,K2: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K2 ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) ) @ ( gbinomial_real @ A @ K2 ) ) ) ).

% gbinomial_factors
thf(fact_7368_gbinomial__factors,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) ) @ ( gbinomial_rat @ A @ K2 ) ) ) ).

% gbinomial_factors
thf(fact_7369_gbinomial__rec,axiom,
    ! [A: complex,K2: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) )
      = ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_rec
thf(fact_7370_gbinomial__rec,axiom,
    ! [A: real,K2: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K2 ) )
      = ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_rec
thf(fact_7371_gbinomial__rec,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) )
      = ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_rec
thf(fact_7372_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_7373_gbinomial__index__swap,axiom,
    ! [K2: nat,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K2 ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_7374_gbinomial__index__swap,axiom,
    ! [K2: nat,N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ K2 ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_7375_gbinomial__index__swap,axiom,
    ! [K2: nat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ K2 ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_7376_gbinomial__negated__upper,axiom,
    ( gbinomial_complex
    = ( ^ [A5: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A5 ) @ one_one_complex ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_7377_gbinomial__negated__upper,axiom,
    ( gbinomial_real
    = ( ^ [A5: real,K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A5 ) @ one_one_real ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_7378_gbinomial__negated__upper,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A5 ) @ one_one_rat ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_7379_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K2: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K2 ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K2 ) )
      = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7380_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K2: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K2 ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K2 ) )
      = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7381_pochhammer__absorb__comp,axiom,
    ! [R2: int,K2: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K2 ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K2 ) )
      = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7382_pochhammer__absorb__comp,axiom,
    ! [R2: real,K2: nat] :
      ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K2 ) )
      = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7383_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K2: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K2 ) )
      = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7384_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).

% pochhammer_same
thf(fact_7385_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).

% pochhammer_same
thf(fact_7386_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% pochhammer_same
thf(fact_7387_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% pochhammer_same
thf(fact_7388_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% pochhammer_same
thf(fact_7389_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_less_lsqrt
thf(fact_7390_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_7391_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% sqrt_even_pow2
thf(fact_7392_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_7393_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_7394_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_7395_ln__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( sqrt @ X ) )
        = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_7396_cos__30,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_30
thf(fact_7397_sin__60,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_60
thf(fact_7398_gbinomial__minus,axiom,
    ! [A: complex,K2: nat] :
      ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K2 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) ) ) ).

% gbinomial_minus
thf(fact_7399_gbinomial__minus,axiom,
    ! [A: real,K2: nat] :
      ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K2 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) ) ) ).

% gbinomial_minus
thf(fact_7400_gbinomial__minus,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) ) ) ).

% gbinomial_minus
thf(fact_7401_complex__norm,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y ) )
      = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_norm
thf(fact_7402_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_complex @ A @ K2 )
        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K2 ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_7403_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_real @ A @ K2 )
        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K2 ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_7404_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_rat @ A @ K2 )
        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K2 ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_7405_pochhammer__minus,axiom,
    ! [B: complex,K2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K2 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) ) ) ).

% pochhammer_minus
thf(fact_7406_pochhammer__minus,axiom,
    ! [B: code_integer,K2: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K2 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K2 ) ) @ one_one_Code_integer ) @ K2 ) ) ) ).

% pochhammer_minus
thf(fact_7407_pochhammer__minus,axiom,
    ! [B: int,K2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K2 ) ) @ one_one_int ) @ K2 ) ) ) ).

% pochhammer_minus
thf(fact_7408_pochhammer__minus,axiom,
    ! [B: real,K2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K2 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) ) ) ).

% pochhammer_minus
thf(fact_7409_pochhammer__minus,axiom,
    ! [B: rat,K2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) ) ) ).

% pochhammer_minus
thf(fact_7410_pochhammer__minus_H,axiom,
    ! [B: complex,K2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K2 ) ) ) ).

% pochhammer_minus'
thf(fact_7411_pochhammer__minus_H,axiom,
    ! [B: code_integer,K2: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K2 ) ) @ one_one_Code_integer ) @ K2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K2 ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K2 ) ) ) ).

% pochhammer_minus'
thf(fact_7412_pochhammer__minus_H,axiom,
    ! [B: int,K2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K2 ) ) @ one_one_int ) @ K2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K2 ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K2 ) ) ) ).

% pochhammer_minus'
thf(fact_7413_pochhammer__minus_H,axiom,
    ! [B: real,K2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K2 ) ) ) ).

% pochhammer_minus'
thf(fact_7414_pochhammer__minus_H,axiom,
    ! [B: rat,K2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K2 ) ) ) ).

% pochhammer_minus'
thf(fact_7415_arsinh__real__aux,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_7416_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_7417_real__sqrt__power__even,axiom,
    ! [N: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( sqrt @ X ) @ N )
          = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_7418_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_7419_tan__30,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).

% tan_30
thf(fact_7420_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K2 )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_7421_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K2 )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_7422_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K2 )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_7423_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_7424_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_7425_arcosh__real__def,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( arcosh_real @ X )
        = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_7426_cos__arctan,axiom,
    ! [X: real] :
      ( ( cos_real @ ( arctan @ X ) )
      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_arctan
thf(fact_7427_sin__arctan,axiom,
    ! [X: real] :
      ( ( sin_real @ ( arctan @ X ) )
      = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arctan
thf(fact_7428_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_complex @ A @ K2 )
        = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_7429_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_real @ A @ K2 )
        = ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_7430_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_rat @ A @ K2 )
        = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_7431_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_7432_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
     => ( ( sin_real @ X )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_7433_arctan__half,axiom,
    ( arctan
    = ( ^ [X4: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X4 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_7434_fact__double,axiom,
    ! [N: nat] :
      ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).

% fact_double
thf(fact_7435_fact__double,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_double
thf(fact_7436_fact__double,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_double
thf(fact_7437_gbinomial__code,axiom,
    ( gbinomial_complex
    = ( ^ [A5: complex,K3: nat] :
          ( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
          @ ( divide1717551699836669952omplex
            @ ( set_fo1517530859248394432omplex
              @ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A5 @ ( semiri8010041392384452111omplex @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_complex )
            @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7438_gbinomial__code,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K3: nat] :
          ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
          @ ( divide_divide_rat
            @ ( set_fo1949268297981939178at_rat
              @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_rat )
            @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7439_gbinomial__code,axiom,
    ( gbinomial_real
    = ( ^ [A5: real,K3: nat] :
          ( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
          @ ( divide_divide_real
            @ ( set_fo3111899725591712190t_real
              @ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A5 @ ( semiri5074537144036343181t_real @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_real )
            @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7440_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,N: nat] :
      ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [K3: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_7441_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K3: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_7442_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K3: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_7443_pochhammer__code,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A5: complex,N3: nat] :
          ( if_complex @ ( N3 = zero_zero_nat ) @ one_one_complex
          @ ( set_fo1517530859248394432omplex
            @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A5 @ ( semiri8010041392384452111omplex @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_complex ) ) ) ) ).

% pochhammer_code
thf(fact_7444_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N3: nat] :
          ( if_int @ ( N3 = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_7445_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A5: real,N3: nat] :
          ( if_real @ ( N3 = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_7446_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N3: nat] :
          ( if_rat @ ( N3 = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_7447_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N3: nat] :
          ( if_nat @ ( N3 = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_7448_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_7449_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M2 @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_7450_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M2 @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_7451_gbinomial__partial__row__sum,axiom,
    ! [A: real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M2 @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_7452_atMost__iff,axiom,
    ! [I: real,K2: real] :
      ( ( member_real @ I @ ( set_ord_atMost_real @ K2 ) )
      = ( ord_less_eq_real @ I @ K2 ) ) ).

% atMost_iff
thf(fact_7453_atMost__iff,axiom,
    ! [I: set_nat,K2: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K2 ) )
      = ( ord_less_eq_set_nat @ I @ K2 ) ) ).

% atMost_iff
thf(fact_7454_atMost__iff,axiom,
    ! [I: rat,K2: rat] :
      ( ( member_rat @ I @ ( set_ord_atMost_rat @ K2 ) )
      = ( ord_less_eq_rat @ I @ K2 ) ) ).

% atMost_iff
thf(fact_7455_atMost__iff,axiom,
    ! [I: num,K2: num] :
      ( ( member_num @ I @ ( set_ord_atMost_num @ K2 ) )
      = ( ord_less_eq_num @ I @ K2 ) ) ).

% atMost_iff
thf(fact_7456_atMost__iff,axiom,
    ! [I: int,K2: int] :
      ( ( member_int @ I @ ( set_ord_atMost_int @ K2 ) )
      = ( ord_less_eq_int @ I @ K2 ) ) ).

% atMost_iff
thf(fact_7457_atMost__iff,axiom,
    ! [I: nat,K2: nat] :
      ( ( member_nat @ I @ ( set_ord_atMost_nat @ K2 ) )
      = ( ord_less_eq_nat @ I @ K2 ) ) ).

% atMost_iff
thf(fact_7458_prod_Oneutral__const,axiom,
    ! [A4: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [Uu3: nat] : one_one_nat
        @ A4 )
      = one_one_nat ) ).

% prod.neutral_const
thf(fact_7459_prod_Oneutral__const,axiom,
    ! [A4: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [Uu3: nat] : one_one_int
        @ A4 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_7460_prod_Oneutral__const,axiom,
    ! [A4: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [Uu3: int] : one_one_int
        @ A4 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_7461_prod_Oempty,axiom,
    ! [G: nat > complex] :
      ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
      = one_one_complex ) ).

% prod.empty
thf(fact_7462_prod_Oempty,axiom,
    ! [G: nat > real] :
      ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
      = one_one_real ) ).

% prod.empty
thf(fact_7463_prod_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
      = one_one_rat ) ).

% prod.empty
thf(fact_7464_prod_Oempty,axiom,
    ! [G: int > complex] :
      ( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
      = one_one_complex ) ).

% prod.empty
thf(fact_7465_prod_Oempty,axiom,
    ! [G: int > real] :
      ( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
      = one_one_real ) ).

% prod.empty
thf(fact_7466_prod_Oempty,axiom,
    ! [G: int > rat] :
      ( ( groups1072433553688619179nt_rat @ G @ bot_bot_set_int )
      = one_one_rat ) ).

% prod.empty
thf(fact_7467_prod_Oempty,axiom,
    ! [G: int > nat] :
      ( ( groups1707563613775114915nt_nat @ G @ bot_bot_set_int )
      = one_one_nat ) ).

% prod.empty
thf(fact_7468_prod_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
      = one_one_complex ) ).

% prod.empty
thf(fact_7469_prod_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
      = one_one_real ) ).

% prod.empty
thf(fact_7470_prod_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
      = one_one_rat ) ).

% prod.empty
thf(fact_7471_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: nat > complex] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( groups6464643781859351333omplex @ G @ A4 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_7472_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: complex > complex] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups3708469109370488835omplex @ G @ A4 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_7473_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: nat > real] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( groups129246275422532515t_real @ G @ A4 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_7474_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups766887009212190081x_real @ G @ A4 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_7475_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( groups73079841787564623at_rat @ G @ A4 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_7476_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups225925009352817453ex_rat @ G @ A4 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_7477_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups861055069439313189ex_nat @ G @ A4 )
        = one_one_nat ) ) ).

% prod.infinite
thf(fact_7478_prod_Oinfinite,axiom,
    ! [A4: set_complex,G: complex > int] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups858564598930262913ex_int @ G @ A4 )
        = one_one_int ) ) ).

% prod.infinite
thf(fact_7479_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: nat > nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( groups708209901874060359at_nat @ G @ A4 )
        = one_one_nat ) ) ).

% prod.infinite
thf(fact_7480_prod_Oinfinite,axiom,
    ! [A4: set_nat,G: nat > int] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( groups705719431365010083at_int @ G @ A4 )
        = one_one_int ) ) ).

% prod.infinite
thf(fact_7481_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_7482_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X ) @ ( set_ord_atMost_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_7483_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X ) @ ( set_ord_atMost_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_7484_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_7485_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_7486_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7487_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7488_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7489_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_7490_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7491_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7492_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7493_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_7494_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_7495_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_7496_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7497_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7498_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7499_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_7500_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7501_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7502_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7503_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_7504_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_7505_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_7506_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_7507_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H2: rat,H3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
      = ( ~ ( ord_less_eq_rat @ L @ H2 )
        | ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7508_Icc__subset__Iic__iff,axiom,
    ! [L: num,H2: num,H3: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
      = ( ~ ( ord_less_eq_num @ L @ H2 )
        | ( ord_less_eq_num @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_7509_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_7510_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_7511_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_7512_sum_OatMost__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atMost_Suc
thf(fact_7513_sum_OatMost__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atMost_Suc
thf(fact_7514_sum_OatMost__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atMost_Suc
thf(fact_7515_sum_OatMost__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atMost_Suc
thf(fact_7516_prod_OlessThan__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% prod.lessThan_Suc
thf(fact_7517_prod_OlessThan__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% prod.lessThan_Suc
thf(fact_7518_prod_OlessThan__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% prod.lessThan_Suc
thf(fact_7519_prod_OlessThan__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% prod.lessThan_Suc
thf(fact_7520_prod_OatMost__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_7521_prod_OatMost__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_7522_prod_OatMost__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_7523_prod_OatMost__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_7524_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7525_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7526_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7527_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7528_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_7529_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > complex,A4: set_nat] :
      ( ( ( groups6464643781859351333omplex @ G @ A4 )
       != one_one_complex )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ( ( G @ A3 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7530_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > complex,A4: set_int] :
      ( ( ( groups7440179247065528705omplex @ G @ A4 )
       != one_one_complex )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ( ( G @ A3 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7531_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > complex,A4: set_real] :
      ( ( ( groups713298508707869441omplex @ G @ A4 )
       != one_one_complex )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ( ( G @ A3 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7532_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > complex,A4: set_complex] :
      ( ( ( groups3708469109370488835omplex @ G @ A4 )
       != one_one_complex )
     => ~ ! [A3: complex] :
            ( ( member_complex @ A3 @ A4 )
           => ( ( G @ A3 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7533_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > real,A4: set_nat] :
      ( ( ( groups129246275422532515t_real @ G @ A4 )
       != one_one_real )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7534_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A4: set_int] :
      ( ( ( groups2316167850115554303t_real @ G @ A4 )
       != one_one_real )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7535_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A4: set_real] :
      ( ( ( groups1681761925125756287l_real @ G @ A4 )
       != one_one_real )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7536_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > real,A4: set_complex] :
      ( ( ( groups766887009212190081x_real @ G @ A4 )
       != one_one_real )
     => ~ ! [A3: complex] :
            ( ( member_complex @ A3 @ A4 )
           => ( ( G @ A3 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7537_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A4: set_nat] :
      ( ( ( groups73079841787564623at_rat @ G @ A4 )
       != one_one_rat )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ( ( G @ A3 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7538_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > rat,A4: set_int] :
      ( ( ( groups1072433553688619179nt_rat @ G @ A4 )
       != one_one_rat )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ( ( G @ A3 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_7539_prod_Oneutral,axiom,
    ! [A4: set_nat,G: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ( G @ X5 )
            = one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ A4 )
        = one_one_nat ) ) ).

% prod.neutral
thf(fact_7540_prod_Oneutral,axiom,
    ! [A4: set_nat,G: nat > int] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ( G @ X5 )
            = one_one_int ) )
     => ( ( groups705719431365010083at_int @ G @ A4 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_7541_prod_Oneutral,axiom,
    ! [A4: set_int,G: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ( G @ X5 )
            = one_one_int ) )
     => ( ( groups1705073143266064639nt_int @ G @ A4 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_7542_prod_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A4: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [X4: nat] : ( times_times_nat @ ( G @ X4 ) @ ( H2 @ X4 ) )
        @ A4 )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A4 ) @ ( groups708209901874060359at_nat @ H2 @ A4 ) ) ) ).

% prod.distrib
thf(fact_7543_prod_Odistrib,axiom,
    ! [G: nat > int,H2: nat > int,A4: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( times_times_int @ ( G @ X4 ) @ ( H2 @ X4 ) )
        @ A4 )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ A4 ) @ ( groups705719431365010083at_int @ H2 @ A4 ) ) ) ).

% prod.distrib
thf(fact_7544_prod_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A4: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( times_times_int @ ( G @ X4 ) @ ( H2 @ X4 ) )
        @ A4 )
      = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A4 ) @ ( groups1705073143266064639nt_int @ H2 @ A4 ) ) ) ).

% prod.distrib
thf(fact_7545_prod__power__distrib,axiom,
    ! [F: nat > nat,A4: set_nat,N: nat] :
      ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ N )
      = ( groups708209901874060359at_nat
        @ ^ [X4: nat] : ( power_power_nat @ ( F @ X4 ) @ N )
        @ A4 ) ) ).

% prod_power_distrib
thf(fact_7546_prod__power__distrib,axiom,
    ! [F: nat > int,A4: set_nat,N: nat] :
      ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A4 ) @ N )
      = ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( power_power_int @ ( F @ X4 ) @ N )
        @ A4 ) ) ).

% prod_power_distrib
thf(fact_7547_prod__power__distrib,axiom,
    ! [F: int > int,A4: set_int,N: nat] :
      ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ N )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( power_power_int @ ( F @ X4 ) @ N )
        @ A4 ) ) ).

% prod_power_distrib
thf(fact_7548_mod__prod__eq,axiom,
    ! [F: nat > nat,A: nat,A4: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ A ) ) ).

% mod_prod_eq
thf(fact_7549_mod__prod__eq,axiom,
    ! [F: nat > int,A: int,A4: set_nat] :
      ( ( modulo_modulo_int
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A4 ) @ A ) ) ).

% mod_prod_eq
thf(fact_7550_mod__prod__eq,axiom,
    ! [F: int > int,A: int,A4: set_int] :
      ( ( modulo_modulo_int
        @ ( groups1705073143266064639nt_int
          @ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ A ) ) ).

% mod_prod_eq
thf(fact_7551_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_7552_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_7553_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_7554_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_7555_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [J3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.nested_swap'
thf(fact_7556_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > int,N: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( groups705719431365010083at_int @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.nested_swap'
thf(fact_7557_atMost__def,axiom,
    ( set_ord_atMost_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X4: real] : ( ord_less_eq_real @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_7558_atMost__def,axiom,
    ( set_or4236626031148496127et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_7559_atMost__def,axiom,
    ( set_ord_atMost_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X4: rat] : ( ord_less_eq_rat @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_7560_atMost__def,axiom,
    ( set_ord_atMost_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X4: num] : ( ord_less_eq_num @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_7561_atMost__def,axiom,
    ( set_ord_atMost_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X4: int] : ( ord_less_eq_int @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_7562_atMost__def,axiom,
    ( set_ord_atMost_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_7563_prod__mono,axiom,
    ! [A4: set_nat,F: nat > real,G: nat > real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7564_prod__mono,axiom,
    ! [A4: set_int,F: int > real,G: int > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7565_prod__mono,axiom,
    ! [A4: set_real,F: real > real,G: real > real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7566_prod__mono,axiom,
    ! [A4: set_complex,F: complex > real,G: complex > real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
            & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7567_prod__mono,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7568_prod__mono,axiom,
    ! [A4: set_int,F: int > rat,G: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7569_prod__mono,axiom,
    ! [A4: set_real,F: real > rat,G: real > rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7570_prod__mono,axiom,
    ! [A4: set_complex,F: complex > rat,G: complex > rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7571_prod__mono,axiom,
    ! [A4: set_int,F: int > nat,G: int > nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A4 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7572_prod__mono,axiom,
    ! [A4: set_real,F: real > nat,G: real > nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ A4 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_7573_prod__nonneg,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).

% prod_nonneg
thf(fact_7574_prod__nonneg,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).

% prod_nonneg
thf(fact_7575_prod__nonneg,axiom,
    ! [A4: set_int,F: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).

% prod_nonneg
thf(fact_7576_prod__pos,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).

% prod_pos
thf(fact_7577_prod__pos,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).

% prod_pos
thf(fact_7578_prod__pos,axiom,
    ! [A4: set_int,F: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).

% prod_pos
thf(fact_7579_prod__ge__1,axiom,
    ! [A4: set_nat,F: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7580_prod__ge__1,axiom,
    ! [A4: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7581_prod__ge__1,axiom,
    ! [A4: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7582_prod__ge__1,axiom,
    ! [A4: set_complex,F: complex > real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7583_prod__ge__1,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7584_prod__ge__1,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7585_prod__ge__1,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7586_prod__ge__1,axiom,
    ! [A4: set_complex,F: complex > rat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7587_prod__ge__1,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7588_prod__ge__1,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_7589_prod__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B: nat] :
      ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A5: nat] : ( times_times_complex @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_complex ) ) ).

% prod_atLeastAtMost_code
thf(fact_7590_prod__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A5: nat] : ( times_times_real @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_real ) ) ).

% prod_atLeastAtMost_code
thf(fact_7591_prod__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A5: nat] : ( times_times_rat @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_rat ) ) ).

% prod_atLeastAtMost_code
thf(fact_7592_prod__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A5: nat] : ( times_times_nat @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_nat ) ) ).

% prod_atLeastAtMost_code
thf(fact_7593_prod__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A5: nat] : ( times_times_int @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_int ) ) ).

% prod_atLeastAtMost_code
thf(fact_7594_prod_OatMost__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.atMost_shift
thf(fact_7595_prod_OatMost__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.atMost_shift
thf(fact_7596_prod_OatMost__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.atMost_shift
thf(fact_7597_prod_OatMost__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.atMost_shift
thf(fact_7598_lessThan__Suc__atMost,axiom,
    ! [K2: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
      = ( set_ord_atMost_nat @ K2 ) ) ).

% lessThan_Suc_atMost
thf(fact_7599_prod_Ointer__filter,axiom,
    ! [A4: set_int,G: int > complex,P: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups7440179247065528705omplex @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( groups7440179247065528705omplex
          @ ^ [X4: int] : ( if_complex @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_complex )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_7600_prod_Ointer__filter,axiom,
    ! [A4: set_real,G: real > complex,P: real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups713298508707869441omplex @ G
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( groups713298508707869441omplex
          @ ^ [X4: real] : ( if_complex @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_complex )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_7601_prod_Ointer__filter,axiom,
    ! [A4: set_nat,G: nat > complex,P: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups6464643781859351333omplex @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( groups6464643781859351333omplex
          @ ^ [X4: nat] : ( if_complex @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_complex )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_7602_prod_Ointer__filter,axiom,
    ! [A4: set_complex,G: complex > complex,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups3708469109370488835omplex @ G
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( groups3708469109370488835omplex
          @ ^ [X4: complex] : ( if_complex @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_complex )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_7603_prod_Ointer__filter,axiom,
    ! [A4: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups2316167850115554303t_real @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( groups2316167850115554303t_real
          @ ^ [X4: int] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_real )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_7604_prod_Ointer__filter,axiom,
    ! [A4: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1681761925125756287l_real @ G
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( groups1681761925125756287l_real
          @ ^ [X4: real] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_real )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_7605_prod_Ointer__filter,axiom,
    ! [A4: set_nat,G: nat > real,P: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups129246275422532515t_real @ G
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( groups129246275422532515t_real
          @ ^ [X4: nat] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_real )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_7606_prod_Ointer__filter,axiom,
    ! [A4: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups766887009212190081x_real @ G
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( groups766887009212190081x_real
          @ ^ [X4: complex] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_real )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_7607_prod_Ointer__filter,axiom,
    ! [A4: set_int,G: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups1072433553688619179nt_rat @ G
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( groups1072433553688619179nt_rat
          @ ^ [X4: int] : ( if_rat @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_rat )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_7608_prod_Ointer__filter,axiom,
    ! [A4: set_real,G: real > rat,P: real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups4061424788464935467al_rat @ G
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A4 )
                & ( P @ X4 ) ) ) )
        = ( groups4061424788464935467al_rat
          @ ^ [X4: real] : ( if_rat @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_rat )
          @ A4 ) ) ) ).

% prod.inter_filter
thf(fact_7609_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M2: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_7610_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > int,M2: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_7611_power__sum,axiom,
    ! [C: real,F: nat > nat,A4: set_nat] :
      ( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups129246275422532515t_real
        @ ^ [A5: nat] : ( power_power_real @ C @ ( F @ A5 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_7612_power__sum,axiom,
    ! [C: complex,F: nat > nat,A4: set_nat] :
      ( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups6464643781859351333omplex
        @ ^ [A5: nat] : ( power_power_complex @ C @ ( F @ A5 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_7613_power__sum,axiom,
    ! [C: nat,F: nat > nat,A4: set_nat] :
      ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups708209901874060359at_nat
        @ ^ [A5: nat] : ( power_power_nat @ C @ ( F @ A5 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_7614_power__sum,axiom,
    ! [C: int,F: nat > nat,A4: set_nat] :
      ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups705719431365010083at_int
        @ ^ [A5: nat] : ( power_power_int @ C @ ( F @ A5 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_7615_power__sum,axiom,
    ! [C: int,F: int > nat,A4: set_int] :
      ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [A5: int] : ( power_power_int @ C @ ( F @ A5 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_7616_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M2: nat,K2: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_7617_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > int,M2: nat,K2: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_7618_prod__le__1,axiom,
    ! [A4: set_nat,F: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_7619_prod__le__1,axiom,
    ! [A4: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_7620_prod__le__1,axiom,
    ! [A4: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_7621_prod__le__1,axiom,
    ! [A4: set_complex,F: complex > real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_7622_prod__le__1,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
            & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_7623_prod__le__1,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
            & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_7624_prod__le__1,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
            & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_7625_prod__le__1,axiom,
    ! [A4: set_complex,F: complex > rat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
            & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_7626_prod__le__1,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) )
            & ( ord_less_eq_nat @ ( F @ X5 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_7627_prod__le__1,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) )
            & ( ord_less_eq_nat @ ( F @ X5 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_7628_prod_Orelated,axiom,
    ! [R: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R @ one_one_complex @ one_one_complex )
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X22 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups6464643781859351333omplex @ H2 @ S3 ) @ ( groups6464643781859351333omplex @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7629_prod_Orelated,axiom,
    ! [R: complex > complex > $o,S3: set_complex,H2: complex > complex,G: complex > complex] :
      ( ( R @ one_one_complex @ one_one_complex )
     => ( ! [X15: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X22 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups3708469109370488835omplex @ H2 @ S3 ) @ ( groups3708469109370488835omplex @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7630_prod_Orelated,axiom,
    ! [R: real > real > $o,S3: set_nat,H2: nat > real,G: nat > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X22 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups129246275422532515t_real @ H2 @ S3 ) @ ( groups129246275422532515t_real @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7631_prod_Orelated,axiom,
    ! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( R @ one_one_real @ one_one_real )
     => ( ! [X15: real,Y15: real,X22: real,Y23: real] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X22 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups766887009212190081x_real @ H2 @ S3 ) @ ( groups766887009212190081x_real @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7632_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X22 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups73079841787564623at_rat @ H2 @ S3 ) @ ( groups73079841787564623at_rat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7633_prod_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R @ one_one_rat @ one_one_rat )
     => ( ! [X15: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X22 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups225925009352817453ex_rat @ H2 @ S3 ) @ ( groups225925009352817453ex_rat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7634_prod_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R @ one_one_nat @ one_one_nat )
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_nat @ X15 @ Y15 ) @ ( times_times_nat @ X22 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups861055069439313189ex_nat @ H2 @ S3 ) @ ( groups861055069439313189ex_nat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7635_prod_Orelated,axiom,
    ! [R: int > int > $o,S3: set_complex,H2: complex > int,G: complex > int] :
      ( ( R @ one_one_int @ one_one_int )
     => ( ! [X15: int,Y15: int,X22: int,Y23: int] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_int @ X15 @ Y15 ) @ ( times_times_int @ X22 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups858564598930262913ex_int @ H2 @ S3 ) @ ( groups858564598930262913ex_int @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7636_prod_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_nat,H2: nat > nat,G: nat > nat] :
      ( ( R @ one_one_nat @ one_one_nat )
     => ( ! [X15: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_nat @ X15 @ Y15 ) @ ( times_times_nat @ X22 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups708209901874060359at_nat @ H2 @ S3 ) @ ( groups708209901874060359at_nat @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7637_prod_Orelated,axiom,
    ! [R: int > int > $o,S3: set_nat,H2: nat > int,G: nat > int] :
      ( ( R @ one_one_int @ one_one_int )
     => ( ! [X15: int,Y15: int,X22: int,Y23: int] :
            ( ( ( R @ X15 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( times_times_int @ X15 @ Y15 ) @ ( times_times_int @ X22 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups705719431365010083at_int @ H2 @ S3 ) @ ( groups705719431365010083at_int @ G @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_7638_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_int,S3: set_int,I: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups7440179247065528705omplex @ G @ S3 )
                        = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7639_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_real,S3: set_int,I: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups7440179247065528705omplex @ G @ S3 )
                        = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7640_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_int,S3: set_real,I: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups713298508707869441omplex @ G @ S3 )
                        = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7641_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_real,S3: set_real,I: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups713298508707869441omplex @ G @ S3 )
                        = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7642_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_complex,S3: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > complex,H2: complex > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: complex] :
                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( ( member_complex @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups7440179247065528705omplex @ G @ S3 )
                        = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7643_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_complex,S3: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > complex,H2: complex > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: complex] :
                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                   => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( ( member_complex @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups713298508707869441omplex @ G @ S3 )
                        = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7644_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T4: set_int,S3: set_complex,I: int > complex,J: complex > int,T3: set_int,G: complex > complex,H2: int > complex] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S3 @ S4 ) ) )
               => ( ! [A3: complex] :
                      ( ( member_complex @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( ( member_complex @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3708469109370488835omplex @ G @ S3 )
                        = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7645_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T4: set_real,S3: set_complex,I: real > complex,J: complex > real,T3: set_real,G: complex > complex,H2: real > complex] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
               => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
                   => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S3 @ S4 ) ) )
               => ( ! [A3: complex] :
                      ( ( member_complex @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( ( member_complex @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3708469109370488835omplex @ G @ S3 )
                        = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7646_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T4: set_complex,S3: set_complex,I: complex > complex,J: complex > complex,T3: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
               => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: complex] :
                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                   => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S3 @ S4 ) ) )
               => ( ! [A3: complex] :
                      ( ( member_complex @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_complex ) )
                 => ( ! [B3: complex] :
                        ( ( member_complex @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = one_one_complex ) )
                   => ( ! [A3: complex] :
                          ( ( member_complex @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3708469109370488835omplex @ G @ S3 )
                        = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7647_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_int,S3: set_int,I: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S4 )
                     => ( ( G @ A3 )
                        = one_one_real ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T4 )
                       => ( ( H2 @ B3 )
                          = one_one_real ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S3 )
                         => ( ( H2 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups2316167850115554303t_real @ G @ S3 )
                        = ( groups2316167850115554303t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_7648_prod_Oin__pairs__0,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.in_pairs_0
thf(fact_7649_prod_Oin__pairs__0,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.in_pairs_0
thf(fact_7650_prod_Oin__pairs__0,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.in_pairs_0
thf(fact_7651_prod_Oin__pairs__0,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.in_pairs_0
thf(fact_7652_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: real > complex] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups713298508707869441omplex @ G
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = one_one_complex ) ) ) )
        = ( groups713298508707869441omplex @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7653_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups3708469109370488835omplex @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = one_one_complex ) ) ) )
        = ( groups3708469109370488835omplex @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7654_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1681761925125756287l_real @ G
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = one_one_real ) ) ) )
        = ( groups1681761925125756287l_real @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7655_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups766887009212190081x_real @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = one_one_real ) ) ) )
        = ( groups766887009212190081x_real @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7656_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups4061424788464935467al_rat @ G
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = one_one_rat ) ) ) )
        = ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7657_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups225925009352817453ex_rat @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = one_one_rat ) ) ) )
        = ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7658_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups4696554848551431203al_nat @ G
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = one_one_nat ) ) ) )
        = ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7659_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups861055069439313189ex_nat @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = one_one_nat ) ) ) )
        = ( groups861055069439313189ex_nat @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7660_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G: real > int] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups4694064378042380927al_int @ G
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G @ X4 )
                  = one_one_int ) ) ) )
        = ( groups4694064378042380927al_int @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7661_prod_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups858564598930262913ex_int @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G @ X4 )
                  = one_one_int ) ) ) )
        = ( groups858564598930262913ex_int @ G @ A4 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_7662_prod_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.nat_diff_reindex
thf(fact_7663_prod_Onat__diff__reindex,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.nat_diff_reindex
thf(fact_7664_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_7665_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
      = ( ord_less_num @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_7666_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_7667_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_7668_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_7669_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N: nat,M2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_7670_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > int,N: nat,M2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_7671_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > complex,H2: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups6464643781859351333omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K2 ) @ one_one_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups6464643781859351333omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7672_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K2 ) @ one_one_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7673_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > rat,H2: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K2 ) @ one_one_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7674_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K2 ) @ one_one_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7675_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K2 ) @ one_one_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7676_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7677_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7678_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: nat > real] :
      ( ( finite_finite_nat @ I6 )
     => ( ( member_nat @ I @ I6 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7679_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 ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I6 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7680_less__1__prod2,axiom,
    ! [I6: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I @ I6 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I6 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7681_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I6 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7682_less__1__prod2,axiom,
    ! [I6: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( member_nat @ I @ I6 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I6 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7683_less__1__prod2,axiom,
    ! [I6: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I @ I6 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I6 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7684_less__1__prod2,axiom,
    ! [I6: set_real,I: real,F: real > int] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I @ I6 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I6 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I2 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7685_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 ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I6 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I2 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_7686_less__1__prod,axiom,
    ! [I6: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_7687_less__1__prod,axiom,
    ! [I6: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ I6 )
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_7688_less__1__prod,axiom,
    ! [I6: set_int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_7689_less__1__prod,axiom,
    ! [I6: set_real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_7690_less__1__prod,axiom,
    ! [I6: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_7691_less__1__prod,axiom,
    ! [I6: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I6 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_7692_less__1__prod,axiom,
    ! [I6: set_int,F: int > rat] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I6 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_7693_less__1__prod,axiom,
    ! [I6: set_real,F: real > rat] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_7694_less__1__prod,axiom,
    ! [I6: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_less_int @ one_one_int @ ( F @ I2 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_7695_less__1__prod,axiom,
    ! [I6: set_real,F: real > int] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_less_int @ one_one_int @ ( F @ I2 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ).

% less_1_prod
thf(fact_7696_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups766887009212190081x_real @ G @ A4 )
          = ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups766887009212190081x_real @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7697_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups225925009352817453ex_rat @ G @ A4 )
          = ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups225925009352817453ex_rat @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7698_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups861055069439313189ex_nat @ G @ A4 )
          = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups861055069439313189ex_nat @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7699_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups858564598930262913ex_int @ G @ A4 )
          = ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups858564598930262913ex_int @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7700_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups129246275422532515t_real @ G @ A4 )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups129246275422532515t_real @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7701_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups73079841787564623at_rat @ G @ A4 )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7702_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups708209901874060359at_nat @ G @ A4 )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7703_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups705719431365010083at_int @ G @ A4 )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups705719431365010083at_int @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7704_prod_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B4 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups1705073143266064639nt_int @ G @ A4 )
          = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ ( minus_minus_set_int @ A4 @ B4 ) ) @ ( groups1705073143266064639nt_int @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_7705_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups7440179247065528705omplex @ G @ T3 )
              = ( groups7440179247065528705omplex @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7706_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups713298508707869441omplex @ G @ T3 )
              = ( groups713298508707869441omplex @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7707_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3708469109370488835omplex @ G @ T3 )
              = ( groups3708469109370488835omplex @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7708_prod_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_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2316167850115554303t_real @ G @ T3 )
              = ( groups2316167850115554303t_real @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7709_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_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ T3 )
              = ( groups1681761925125756287l_real @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7710_prod_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 )
                = one_one_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups766887009212190081x_real @ G @ T3 )
              = ( groups766887009212190081x_real @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7711_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1072433553688619179nt_rat @ G @ T3 )
              = ( groups1072433553688619179nt_rat @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7712_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4061424788464935467al_rat @ G @ T3 )
              = ( groups4061424788464935467al_rat @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7713_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups225925009352817453ex_rat @ G @ T3 )
              = ( groups225925009352817453ex_rat @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7714_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_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1707563613775114915nt_nat @ G @ T3 )
              = ( groups1707563613775114915nt_nat @ H2 @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_7715_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H2: int > complex,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_complex ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups7440179247065528705omplex @ G @ S3 )
              = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7716_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups713298508707869441omplex @ G @ S3 )
              = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7717_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_complex ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3708469109370488835omplex @ G @ S3 )
              = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7718_prod_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_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_real ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2316167850115554303t_real @ G @ S3 )
              = ( groups2316167850115554303t_real @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7719_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_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ S3 )
              = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7720_prod_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 )
                = one_one_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups766887009212190081x_real @ G @ S3 )
              = ( groups766887009212190081x_real @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7721_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_rat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1072433553688619179nt_rat @ G @ S3 )
              = ( groups1072433553688619179nt_rat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7722_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4061424788464935467al_rat @ G @ S3 )
              = ( groups4061424788464935467al_rat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7723_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups225925009352817453ex_rat @ G @ S3 )
              = ( groups225925009352817453ex_rat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7724_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_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = one_one_nat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1707563613775114915nt_nat @ G @ S3 )
              = ( groups1707563613775114915nt_nat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_7725_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_7726_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_7727_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G @ T3 )
            = ( groups225925009352817453ex_rat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7728_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_7729_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_7730_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups6464643781859351333omplex @ G @ T3 )
            = ( groups6464643781859351333omplex @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7731_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups129246275422532515t_real @ G @ T3 )
            = ( groups129246275422532515t_real @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7732_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ T3 )
            = ( groups73079841787564623at_rat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7733_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups708209901874060359at_nat @ G @ T3 )
            = ( groups708209901874060359at_nat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7734_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups705719431365010083at_int @ G @ T3 )
            = ( groups705719431365010083at_int @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_7735_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_7736_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_7737_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G @ S3 )
            = ( groups225925009352817453ex_rat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7738_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_7739_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_7740_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups6464643781859351333omplex @ G @ S3 )
            = ( groups6464643781859351333omplex @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7741_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups129246275422532515t_real @ G @ S3 )
            = ( groups129246275422532515t_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7742_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ S3 )
            = ( groups73079841787564623at_rat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7743_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups708209901874060359at_nat @ G @ S3 )
            = ( groups708209901874060359at_nat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7744_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups705719431365010083at_int @ G @ S3 )
            = ( groups705719431365010083at_int @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_7745_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups7440179247065528705omplex @ G @ C4 )
                  = ( groups7440179247065528705omplex @ H2 @ C4 ) )
               => ( ( groups7440179247065528705omplex @ G @ A4 )
                  = ( groups7440179247065528705omplex @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7746_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups713298508707869441omplex @ G @ C4 )
                  = ( groups713298508707869441omplex @ H2 @ C4 ) )
               => ( ( groups713298508707869441omplex @ G @ A4 )
                  = ( groups713298508707869441omplex @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7747_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G @ C4 )
                  = ( groups3708469109370488835omplex @ H2 @ C4 ) )
               => ( ( groups3708469109370488835omplex @ G @ A4 )
                  = ( groups3708469109370488835omplex @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7748_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups2316167850115554303t_real @ G @ C4 )
                  = ( groups2316167850115554303t_real @ H2 @ C4 ) )
               => ( ( groups2316167850115554303t_real @ G @ A4 )
                  = ( groups2316167850115554303t_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7749_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) )
               => ( ( groups1681761925125756287l_real @ G @ A4 )
                  = ( groups1681761925125756287l_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7750_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G @ C4 )
                  = ( groups766887009212190081x_real @ H2 @ C4 ) )
               => ( ( groups766887009212190081x_real @ G @ A4 )
                  = ( groups766887009212190081x_real @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7751_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups1072433553688619179nt_rat @ G @ C4 )
                  = ( groups1072433553688619179nt_rat @ H2 @ C4 ) )
               => ( ( groups1072433553688619179nt_rat @ G @ A4 )
                  = ( groups1072433553688619179nt_rat @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7752_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups4061424788464935467al_rat @ G @ C4 )
                  = ( groups4061424788464935467al_rat @ H2 @ C4 ) )
               => ( ( groups4061424788464935467al_rat @ G @ A4 )
                  = ( groups4061424788464935467al_rat @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7753_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups225925009352817453ex_rat @ G @ C4 )
                  = ( groups225925009352817453ex_rat @ H2 @ C4 ) )
               => ( ( groups225925009352817453ex_rat @ G @ A4 )
                  = ( groups225925009352817453ex_rat @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7754_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_nat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_nat ) )
             => ( ( ( groups1707563613775114915nt_nat @ G @ C4 )
                  = ( groups1707563613775114915nt_nat @ H2 @ C4 ) )
               => ( ( groups1707563613775114915nt_nat @ G @ A4 )
                  = ( groups1707563613775114915nt_nat @ H2 @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_7755_prod_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups7440179247065528705omplex @ G @ A4 )
                  = ( groups7440179247065528705omplex @ H2 @ B4 ) )
                = ( ( groups7440179247065528705omplex @ G @ C4 )
                  = ( groups7440179247065528705omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7756_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups713298508707869441omplex @ G @ A4 )
                  = ( groups713298508707869441omplex @ H2 @ B4 ) )
                = ( ( groups713298508707869441omplex @ G @ C4 )
                  = ( groups713298508707869441omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7757_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G @ A4 )
                  = ( groups3708469109370488835omplex @ H2 @ B4 ) )
                = ( ( groups3708469109370488835omplex @ G @ C4 )
                  = ( groups3708469109370488835omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7758_prod_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups2316167850115554303t_real @ G @ A4 )
                  = ( groups2316167850115554303t_real @ H2 @ B4 ) )
                = ( ( groups2316167850115554303t_real @ G @ C4 )
                  = ( groups2316167850115554303t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7759_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ A4 )
                  = ( groups1681761925125756287l_real @ H2 @ B4 ) )
                = ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7760_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G @ A4 )
                  = ( groups766887009212190081x_real @ H2 @ B4 ) )
                = ( ( groups766887009212190081x_real @ G @ C4 )
                  = ( groups766887009212190081x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7761_prod_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups1072433553688619179nt_rat @ G @ A4 )
                  = ( groups1072433553688619179nt_rat @ H2 @ B4 ) )
                = ( ( groups1072433553688619179nt_rat @ G @ C4 )
                  = ( groups1072433553688619179nt_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7762_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B4: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B4 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups4061424788464935467al_rat @ G @ A4 )
                  = ( groups4061424788464935467al_rat @ H2 @ B4 ) )
                = ( ( groups4061424788464935467al_rat @ G @ C4 )
                  = ( groups4061424788464935467al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7763_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B4: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups225925009352817453ex_rat @ G @ A4 )
                  = ( groups225925009352817453ex_rat @ H2 @ B4 ) )
                = ( ( groups225925009352817453ex_rat @ G @ C4 )
                  = ( groups225925009352817453ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7764_prod_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B4: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B4 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_nat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B4 ) )
                 => ( ( H2 @ B3 )
                    = one_one_nat ) )
             => ( ( ( groups1707563613775114915nt_nat @ G @ A4 )
                  = ( groups1707563613775114915nt_nat @ H2 @ B4 ) )
                = ( ( groups1707563613775114915nt_nat @ G @ C4 )
                  = ( groups1707563613775114915nt_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_7765_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7766_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7767_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7768_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_7769_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_real @ ( G @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7770_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_rat @ ( G @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7771_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_nat @ ( G @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7772_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_int @ ( G @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_7773_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_real @ ( G @ M2 ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7774_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_rat @ ( G @ M2 ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7775_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_nat @ ( G @ M2 ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7776_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_int @ ( G @ M2 ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_7777_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_7778_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_7779_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_7780_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_7781_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_real @ ( G @ M2 )
          @ ( groups129246275422532515t_real
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7782_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_rat @ ( G @ M2 )
          @ ( groups73079841787564623at_rat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7783_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_nat @ ( G @ M2 )
          @ ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7784_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_int @ ( G @ M2 )
          @ ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_7785_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_7786_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_7787_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_7788_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_7789_sum__telescope,axiom,
    ! [F: nat > rat,I: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_7790_sum__telescope,axiom,
    ! [F: nat > int,I: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_7791_sum__telescope,axiom,
    ! [F: nat > real,I: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_7792_polyfun__eq__coeffs,axiom,
    ! [C: nat > complex,N: nat,D: nat > complex] :
      ( ( ! [X4: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( D @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N ) ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N )
           => ( ( C @ I3 )
              = ( D @ I3 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_7793_polyfun__eq__coeffs,axiom,
    ! [C: nat > real,N: nat,D: nat > real] :
      ( ( ! [X4: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( D @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N ) ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N )
           => ( ( C @ I3 )
              = ( D @ I3 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_7794_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_7795_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_7796_bounded__imp__summable,axiom,
    ! [A: nat > int,B4: int] :
      ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N2 ) ) @ B4 )
       => ( summable_int @ A ) ) ) ).

% bounded_imp_summable
thf(fact_7797_bounded__imp__summable,axiom,
    ! [A: nat > nat,B4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N2 ) ) @ B4 )
       => ( summable_nat @ A ) ) ) ).

% bounded_imp_summable
thf(fact_7798_bounded__imp__summable,axiom,
    ! [A: nat > real,B4: real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N2 ) ) @ B4 )
       => ( summable_real @ A ) ) ) ).

% bounded_imp_summable
thf(fact_7799_fact__prod,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N3: nat] :
          ( semiri1314217659103216013at_int
          @ ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ) ).

% fact_prod
thf(fact_7800_fact__prod,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [N3: nat] :
          ( semiri681578069525770553at_rat
          @ ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ) ).

% fact_prod
thf(fact_7801_fact__prod,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N3: nat] :
          ( semiri5074537144036343181t_real
          @ ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ) ).

% fact_prod
thf(fact_7802_fact__prod,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N3: nat] :
          ( semiri1316708129612266289at_nat
          @ ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ) ).

% fact_prod
thf(fact_7803_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nested_swap'
thf(fact_7804_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( groups6591440286371151544t_real
        @ ^ [J3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nested_swap'
thf(fact_7805_prod__mono__strict,axiom,
    ! [A4: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ A4 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A4 != bot_bot_set_complex )
         => ( ord_less_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7806_prod__mono__strict,axiom,
    ! [A4: set_nat,F: nat > real,G: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ A4 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A4 != bot_bot_set_nat )
         => ( ord_less_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7807_prod__mono__strict,axiom,
    ! [A4: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ A4 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A4 != bot_bot_set_int )
         => ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7808_prod__mono__strict,axiom,
    ! [A4: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ A4 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
              & ( ord_less_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A4 != bot_bot_set_real )
         => ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7809_prod__mono__strict,axiom,
    ! [A4: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A4 != bot_bot_set_complex )
         => ( ord_less_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7810_prod__mono__strict,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A4 != bot_bot_set_nat )
         => ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7811_prod__mono__strict,axiom,
    ! [A4: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A4 != bot_bot_set_int )
         => ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7812_prod__mono__strict,axiom,
    ! [A4: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
              & ( ord_less_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A4 != bot_bot_set_real )
         => ( ord_less_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7813_prod__mono__strict,axiom,
    ! [A4: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ A4 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
              & ( ord_less_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A4 != bot_bot_set_complex )
         => ( ord_less_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7814_prod__mono__strict,axiom,
    ! [A4: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ A4 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
              & ( ord_less_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
       => ( ( A4 != bot_bot_set_int )
         => ( ord_less_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_7815_even__prod__iff,axiom,
    ! [A4: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ A4 )
     => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3455450783089532116nteger @ F @ A4 ) )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7816_even__prod__iff,axiom,
    ! [A4: set_complex,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups8682486955453173170nteger @ F @ A4 ) )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7817_even__prod__iff,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups861055069439313189ex_nat @ F @ A4 ) )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7818_even__prod__iff,axiom,
    ! [A4: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups858564598930262913ex_int @ F @ A4 ) )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7819_even__prod__iff,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups708209901874060359at_nat @ F @ A4 ) )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7820_even__prod__iff,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups705719431365010083at_int @ F @ A4 ) )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7821_even__prod__iff,axiom,
    ! [A4: set_int,F: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups1705073143266064639nt_int @ F @ A4 ) )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_7822_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > real,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7823_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > rat,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7824_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > nat,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7825_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > int,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_7826_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb2 @ Xc )
        = Y )
     => ( ( ( ord_less_nat @ Xb2 @ Xa2 )
         => ( Y = Xc ) )
        & ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
         => ( Y
            = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_7827_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F3: nat > nat > nat,A5: nat,B5: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B5 @ A5 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F3 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B5 @ ( F3 @ A5 @ Acc2 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_7828_norm__prod__diff,axiom,
    ! [I6: set_int,Z: int > real,W: int > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I6 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I6 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups2316167850115554303t_real @ Z @ I6 ) @ ( groups2316167850115554303t_real @ W @ I6 ) ) )
          @ ( groups8778361861064173332t_real
            @ ^ [I3: int] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_7829_norm__prod__diff,axiom,
    ! [I6: set_real,Z: real > real,W: real > real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I6 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I6 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups1681761925125756287l_real @ Z @ I6 ) @ ( groups1681761925125756287l_real @ W @ I6 ) ) )
          @ ( groups8097168146408367636l_real
            @ ^ [I3: real] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_7830_norm__prod__diff,axiom,
    ! [I6: set_complex,Z: complex > real,W: complex > real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I6 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I6 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups766887009212190081x_real @ Z @ I6 ) @ ( groups766887009212190081x_real @ W @ I6 ) ) )
          @ ( groups5808333547571424918x_real
            @ ^ [I3: complex] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_7831_norm__prod__diff,axiom,
    ! [I6: set_Pr1261947904930325089at_nat,Z: product_prod_nat_nat > real,W: product_prod_nat_nat > real] :
      ( ! [I2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I2 @ I6 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I2 @ I6 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups6036352826371341000t_real @ Z @ I6 ) @ ( groups6036352826371341000t_real @ W @ I6 ) ) )
          @ ( groups4567486121110086003t_real
            @ ^ [I3: product_prod_nat_nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_7832_norm__prod__diff,axiom,
    ! [I6: set_int,Z: int > complex,W: int > complex] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I6 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I6 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups7440179247065528705omplex @ Z @ I6 ) @ ( groups7440179247065528705omplex @ W @ I6 ) ) )
          @ ( groups8778361861064173332t_real
            @ ^ [I3: int] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_7833_norm__prod__diff,axiom,
    ! [I6: set_real,Z: real > complex,W: real > complex] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I6 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I6 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups713298508707869441omplex @ Z @ I6 ) @ ( groups713298508707869441omplex @ W @ I6 ) ) )
          @ ( groups8097168146408367636l_real
            @ ^ [I3: real] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_7834_norm__prod__diff,axiom,
    ! [I6: set_complex,Z: complex > complex,W: complex > complex] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I6 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I6 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups3708469109370488835omplex @ Z @ I6 ) @ ( groups3708469109370488835omplex @ W @ I6 ) ) )
          @ ( groups5808333547571424918x_real
            @ ^ [I3: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_7835_norm__prod__diff,axiom,
    ! [I6: set_Pr1261947904930325089at_nat,Z: product_prod_nat_nat > complex,W: product_prod_nat_nat > complex] :
      ( ! [I2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I2 @ I6 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ I2 @ I6 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups8110221916422527690omplex @ Z @ I6 ) @ ( groups8110221916422527690omplex @ W @ I6 ) ) )
          @ ( groups4567486121110086003t_real
            @ ^ [I3: product_prod_nat_nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_7836_norm__prod__diff,axiom,
    ! [I6: set_nat,Z: nat > real,W: nat > real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I6 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I6 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups129246275422532515t_real @ Z @ I6 ) @ ( groups129246275422532515t_real @ W @ I6 ) ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_7837_norm__prod__diff,axiom,
    ! [I6: set_nat,Z: nat > complex,W: nat > complex] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I6 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I2 ) ) @ one_one_real ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I6 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I2 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups6464643781859351333omplex @ Z @ I6 ) @ ( groups6464643781859351333omplex @ W @ I6 ) ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I3 ) @ ( W @ I3 ) ) )
            @ I6 ) ) ) ) ).

% norm_prod_diff
thf(fact_7838_polyfun__eq__0,axiom,
    ! [C: nat > complex,N: nat] :
      ( ( ! [X4: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = zero_zero_complex ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N )
           => ( ( C @ I3 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_0
thf(fact_7839_polyfun__eq__0,axiom,
    ! [C: nat > real,N: nat] :
      ( ( ! [X4: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = zero_zero_real ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N )
           => ( ( C @ I3 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_0
thf(fact_7840_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > complex,N: nat,K2: nat] :
      ( ! [W3: complex] :
          ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ W3 @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_complex )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( C @ K2 )
          = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_7841_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > real,N: nat,K2: nat] :
      ( ! [W3: real] :
          ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ W3 @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_real )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( C @ K2 )
          = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_7842_sum_OatMost__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.atMost_shift
thf(fact_7843_sum_OatMost__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.atMost_shift
thf(fact_7844_sum_OatMost__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.atMost_shift
thf(fact_7845_sum_OatMost__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.atMost_shift
thf(fact_7846_sum__up__index__split,axiom,
    ! [F: nat > rat,M2: nat,N: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_7847_sum__up__index__split,axiom,
    ! [F: nat > int,M2: nat,N: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_7848_sum__up__index__split,axiom,
    ! [F: nat > nat,M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_7849_sum__up__index__split,axiom,
    ! [F: nat > real,M2: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_7850_gbinomial__parallel__sum,axiom,
    ! [A: complex,N: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K3 ) ) @ K3 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ N ) ) ).

% gbinomial_parallel_sum
thf(fact_7851_gbinomial__parallel__sum,axiom,
    ! [A: rat,N: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K3 ) ) @ K3 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ N ) ) ).

% gbinomial_parallel_sum
thf(fact_7852_gbinomial__parallel__sum,axiom,
    ! [A: real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K3 ) ) @ K3 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ N ) ) ).

% gbinomial_parallel_sum
thf(fact_7853_sum_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_7854_sum_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > real,N: nat] :
      ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_7855_fact__eq__fact__times,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri1408675320244567234ct_nat @ M2 )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
          @ ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M2 ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_7856_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: int > real] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ A4 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ A4 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ F @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7857_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > real] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A4 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ F @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7858_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A4 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ F @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7859_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: int > rat] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ F @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7860_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > rat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ F @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7861_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ F @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7862_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > int] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A4 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
           => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ F @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7863_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A4 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
           => ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7864_prod__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B4 @ A4 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ A4 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ F @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7865_prod__mono2,axiom,
    ! [B4: set_nat,A4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ F @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_7866_sum__gp__basic,axiom,
    ! [X: complex,N: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ).

% sum_gp_basic
thf(fact_7867_sum__gp__basic,axiom,
    ! [X: rat,N: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) ) )
      = ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ).

% sum_gp_basic
thf(fact_7868_sum__gp__basic,axiom,
    ! [X: int,N: nat] :
      ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N ) ) )
      = ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ).

% sum_gp_basic
thf(fact_7869_sum__gp__basic,axiom,
    ! [X: real,N: nat] :
      ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ).

% sum_gp_basic
thf(fact_7870_polyfun__finite__roots,axiom,
    ! [C: nat > complex,N: nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] :
              ( ( groups2073611262835488442omplex
                @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
                @ ( set_ord_atMost_nat @ N ) )
              = zero_zero_complex ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N )
            & ( ( C @ I3 )
             != zero_zero_complex ) ) ) ) ).

% polyfun_finite_roots
thf(fact_7871_polyfun__finite__roots,axiom,
    ! [C: nat > real,N: nat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X4: real] :
              ( ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
                @ ( set_ord_atMost_nat @ N ) )
              = zero_zero_real ) ) )
      = ( ? [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N )
            & ( ( C @ I3 )
             != zero_zero_real ) ) ) ) ).

% polyfun_finite_roots
thf(fact_7872_polyfun__roots__finite,axiom,
    ! [C: nat > complex,K2: nat,N: nat] :
      ( ( ( C @ K2 )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z2 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = zero_zero_complex ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_7873_polyfun__roots__finite,axiom,
    ! [C: nat > real,K2: nat,N: nat] :
      ( ( ( C @ K2 )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Z2: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z2 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = zero_zero_real ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_7874_pochhammer__Suc__prod,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7875_pochhammer__Suc__prod,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7876_pochhammer__Suc__prod,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7877_pochhammer__Suc__prod,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_7878_polyfun__linear__factor__root,axiom,
    ! [C: nat > complex,A: complex,N: nat] :
      ( ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_complex )
     => ~ ! [B3: nat > complex] :
            ~ ! [Z5: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z5 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_complex @ ( minus_minus_complex @ Z5 @ A )
                  @ ( groups2073611262835488442omplex
                    @ ^ [I3: nat] : ( times_times_complex @ ( B3 @ I3 ) @ ( power_power_complex @ Z5 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_7879_polyfun__linear__factor__root,axiom,
    ! [C: nat > rat,A: rat,N: nat] :
      ( ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_rat )
     => ~ ! [B3: nat > rat] :
            ~ ! [Z5: rat] :
                ( ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z5 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_rat @ ( minus_minus_rat @ Z5 @ A )
                  @ ( groups2906978787729119204at_rat
                    @ ^ [I3: nat] : ( times_times_rat @ ( B3 @ I3 ) @ ( power_power_rat @ Z5 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_7880_polyfun__linear__factor__root,axiom,
    ! [C: nat > int,A: int,N: nat] :
      ( ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_int )
     => ~ ! [B3: nat > int] :
            ~ ! [Z5: int] :
                ( ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z5 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_int @ ( minus_minus_int @ Z5 @ A )
                  @ ( groups3539618377306564664at_int
                    @ ^ [I3: nat] : ( times_times_int @ ( B3 @ I3 ) @ ( power_power_int @ Z5 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_7881_polyfun__linear__factor__root,axiom,
    ! [C: nat > real,A: real,N: nat] :
      ( ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_real )
     => ~ ! [B3: nat > real] :
            ~ ! [Z5: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z5 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_real @ ( minus_minus_real @ Z5 @ A )
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( B3 @ I3 ) @ ( power_power_real @ Z5 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_7882_polyfun__linear__factor,axiom,
    ! [C: nat > complex,N: nat,A: complex] :
    ? [B3: nat > complex] :
    ! [Z5: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z5 @ I3 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_complex
        @ ( times_times_complex @ ( minus_minus_complex @ Z5 @ A )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( B3 @ I3 ) @ ( power_power_complex @ Z5 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% polyfun_linear_factor
thf(fact_7883_polyfun__linear__factor,axiom,
    ! [C: nat > rat,N: nat,A: rat] :
    ? [B3: nat > rat] :
    ! [Z5: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z5 @ I3 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_rat
        @ ( times_times_rat @ ( minus_minus_rat @ Z5 @ A )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( B3 @ I3 ) @ ( power_power_rat @ Z5 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% polyfun_linear_factor
thf(fact_7884_polyfun__linear__factor,axiom,
    ! [C: nat > int,N: nat,A: int] :
    ? [B3: nat > int] :
    ! [Z5: int] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z5 @ I3 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_int
        @ ( times_times_int @ ( minus_minus_int @ Z5 @ A )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( B3 @ I3 ) @ ( power_power_int @ Z5 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% polyfun_linear_factor
thf(fact_7885_polyfun__linear__factor,axiom,
    ! [C: nat > real,N: nat,A: real] :
    ? [B3: nat > real] :
    ! [Z5: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z5 @ I3 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_real
        @ ( times_times_real @ ( minus_minus_real @ Z5 @ A )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( B3 @ I3 ) @ ( power_power_real @ Z5 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% polyfun_linear_factor
thf(fact_7886_pochhammer__prod__rev,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A5: real,N3: nat] :
          ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N3 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7887_pochhammer__prod__rev,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N3: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N3 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7888_pochhammer__prod__rev,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N3: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N3 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7889_pochhammer__prod__rev,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N3: nat] :
          ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N3 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_7890_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: complex] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_7891_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_7892_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_int @ ( power_power_int @ X @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_7893_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_7894_sum_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.triangle_reindex
thf(fact_7895_sum_Otriangle__reindex,axiom,
    ! [G: nat > nat > real,N: nat] :
      ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.triangle_reindex
thf(fact_7896_fact__div__fact,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) )
        = ( groups708209901874060359at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) ) ).

% fact_div_fact
thf(fact_7897_summable__Cauchy__product,axiom,
    ! [A: nat > complex,B: nat > complex] :
      ( ( summable_real
        @ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( A @ K3 ) ) )
     => ( ( summable_real
          @ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( B @ K3 ) ) )
       => ( summable_complex
          @ ^ [K3: nat] :
              ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
              @ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ).

% summable_Cauchy_product
thf(fact_7898_summable__Cauchy__product,axiom,
    ! [A: nat > real,B: nat > real] :
      ( ( summable_real
        @ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( A @ K3 ) ) )
     => ( ( summable_real
          @ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( B @ K3 ) ) )
       => ( summable_real
          @ ^ [K3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
              @ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ).

% summable_Cauchy_product
thf(fact_7899_Cauchy__product,axiom,
    ! [A: nat > complex,B: nat > complex] :
      ( ( summable_real
        @ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( A @ K3 ) ) )
     => ( ( summable_real
          @ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( B @ K3 ) ) )
       => ( ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B ) )
          = ( suminf_complex
            @ ^ [K3: nat] :
                ( groups2073611262835488442omplex
                @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
                @ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ) ).

% Cauchy_product
thf(fact_7900_Cauchy__product,axiom,
    ! [A: nat > real,B: nat > real] :
      ( ( summable_real
        @ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( A @ K3 ) ) )
     => ( ( summable_real
          @ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( B @ K3 ) ) )
       => ( ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B ) )
          = ( suminf_real
            @ ^ [K3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
                @ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ) ).

% Cauchy_product
thf(fact_7901_prod_Oin__pairs,axiom,
    ! [G: nat > real,M2: nat,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_7902_prod_Oin__pairs,axiom,
    ! [G: nat > rat,M2: nat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_7903_prod_Oin__pairs,axiom,
    ! [G: nat > nat,M2: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_7904_prod_Oin__pairs,axiom,
    ! [G: nat > int,M2: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_7905_sum__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B: nat] :
      ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A5: nat] : ( plus_plus_complex @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_complex ) ) ).

% sum_atLeastAtMost_code
thf(fact_7906_sum__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A5: nat] : ( plus_plus_rat @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_rat ) ) ).

% sum_atLeastAtMost_code
thf(fact_7907_sum__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A5: nat] : ( plus_plus_int @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_7908_sum__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A5: nat] : ( plus_plus_nat @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_7909_sum__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A5: nat] : ( plus_plus_real @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_real ) ) ).

% sum_atLeastAtMost_code
thf(fact_7910_sum_Oin__pairs__0,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.in_pairs_0
thf(fact_7911_sum_Oin__pairs__0,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.in_pairs_0
thf(fact_7912_sum_Oin__pairs__0,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.in_pairs_0
thf(fact_7913_sum_Oin__pairs__0,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.in_pairs_0
thf(fact_7914_polynomial__product,axiom,
    ! [M2: nat,A: nat > complex,N: nat,B: nat > complex,X: complex] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M2 @ I2 )
         => ( ( A @ I2 )
            = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B @ J2 )
              = zero_zero_complex ) )
       => ( ( times_times_complex
            @ ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups2073611262835488442omplex
              @ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups2073611262835488442omplex
            @ ^ [R5: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [K3: nat] : ( times_times_complex @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_complex @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_7915_polynomial__product,axiom,
    ! [M2: nat,A: nat > rat,N: nat,B: nat > rat,X: rat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M2 @ I2 )
         => ( ( A @ I2 )
            = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B @ J2 )
              = zero_zero_rat ) )
       => ( ( times_times_rat
            @ ( groups2906978787729119204at_rat
              @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups2906978787729119204at_rat
              @ ^ [J3: nat] : ( times_times_rat @ ( B @ J3 ) @ ( power_power_rat @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups2906978787729119204at_rat
            @ ^ [R5: nat] :
                ( times_times_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [K3: nat] : ( times_times_rat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_rat @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_7916_polynomial__product,axiom,
    ! [M2: nat,A: nat > int,N: nat,B: nat > int,X: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M2 @ I2 )
         => ( ( A @ I2 )
            = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B @ J2 )
              = zero_zero_int ) )
       => ( ( times_times_int
            @ ( groups3539618377306564664at_int
              @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups3539618377306564664at_int
              @ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3539618377306564664at_int
            @ ^ [R5: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [K3: nat] : ( times_times_int @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_int @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_7917_polynomial__product,axiom,
    ! [M2: nat,A: nat > real,N: nat,B: nat > real,X: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M2 @ I2 )
         => ( ( A @ I2 )
            = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B @ J2 )
              = zero_zero_real ) )
       => ( ( times_times_real
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups6591440286371151544t_real
              @ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups6591440286371151544t_real
            @ ^ [R5: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [K3: nat] : ( times_times_real @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_real @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_7918_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( groups129246275422532515t_real
        @ ^ [I3: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_7919_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_7920_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_7921_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_7922_polyfun__eq__const,axiom,
    ! [C: nat > complex,N: nat,K2: complex] :
      ( ( ! [X4: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = K2 ) )
      = ( ( ( C @ zero_zero_nat )
          = K2 )
        & ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
           => ( ( C @ X4 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
thf(fact_7923_polyfun__eq__const,axiom,
    ! [C: nat > real,N: nat,K2: real] :
      ( ( ! [X4: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = K2 ) )
      = ( ( ( C @ zero_zero_nat )
          = K2 )
        & ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
           => ( ( C @ X4 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
thf(fact_7924_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ M2 ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_7925_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ M2 ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_7926_gbinomial__sum__lower__neg,axiom,
    ! [A: real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ M2 ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ M2 ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_7927_polynomial__product__nat,axiom,
    ! [M2: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M2 @ I2 )
         => ( ( A @ I2 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R5: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_nat @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_7928_Cauchy__product__sums,axiom,
    ! [A: nat > complex,B: nat > complex] :
      ( ( summable_real
        @ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( A @ K3 ) ) )
     => ( ( summable_real
          @ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( B @ K3 ) ) )
       => ( sums_complex
          @ ^ [K3: nat] :
              ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
              @ ( set_ord_atMost_nat @ K3 ) )
          @ ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B ) ) ) ) ) ).

% Cauchy_product_sums
thf(fact_7929_Cauchy__product__sums,axiom,
    ! [A: nat > real,B: nat > real] :
      ( ( summable_real
        @ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( A @ K3 ) ) )
     => ( ( summable_real
          @ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( B @ K3 ) ) )
       => ( sums_real
          @ ^ [K3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( B @ ( minus_minus_nat @ K3 @ I3 ) ) )
              @ ( set_ord_atMost_nat @ K3 ) )
          @ ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B ) ) ) ) ) ).

% Cauchy_product_sums
thf(fact_7930_gbinomial__Suc,axiom,
    ! [A: complex,K2: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K2 ) )
      = ( divide1717551699836669952omplex
        @ ( groups6464643781859351333omplex
          @ ^ [I3: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri5044797733671781792omplex @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_7931_gbinomial__Suc,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K2 ) )
      = ( divide_divide_rat
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri773545260158071498ct_rat @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_7932_gbinomial__Suc,axiom,
    ! [A: real,K2: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K2 ) )
      = ( divide_divide_real
        @ ( groups129246275422532515t_real
          @ ^ [I3: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri2265585572941072030t_real @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_7933_gbinomial__Suc,axiom,
    ! [A: nat,K2: nat] :
      ( ( gbinomial_nat @ A @ ( suc @ K2 ) )
      = ( divide_divide_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri1408675320244567234ct_nat @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_7934_gbinomial__Suc,axiom,
    ! [A: int,K2: nat] :
      ( ( gbinomial_int @ A @ ( suc @ K2 ) )
      = ( divide_divide_int
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri1406184849735516958ct_int @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_7935_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > complex,H2: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K2 ) @ zero_zero_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups2073611262835488442omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7936_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > rat,H2: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K2 ) @ zero_zero_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7937_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K2 ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7938_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K2 ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7939_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K2 ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7940_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: complex,X: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ A ) @ K3 ) @ ( power_power_complex @ X @ K3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_7941_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: rat,X: rat,Y: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ A ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X ) @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_7942_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: real,X: real,Y: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ A ) @ K3 ) @ ( power_power_real @ X @ K3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_7943_root__polyfun,axiom,
    ! [N: nat,Z: int,A: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_int @ Z @ N )
          = A )
        = ( ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( if_int @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I3 = N ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_int ) ) ) ).

% root_polyfun
thf(fact_7944_root__polyfun,axiom,
    ! [N: nat,Z: complex,A: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_complex @ Z @ N )
          = A )
        = ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( if_complex @ ( I3 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I3 = N ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_complex ) ) ) ).

% root_polyfun
thf(fact_7945_root__polyfun,axiom,
    ! [N: nat,Z: rat,A: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_rat @ Z @ N )
          = A )
        = ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( if_rat @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I3 = N ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_rat ) ) ) ).

% root_polyfun
thf(fact_7946_root__polyfun,axiom,
    ! [N: nat,Z: code_integer,A: code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_8256067586552552935nteger @ Z @ N )
          = A )
        = ( ( groups7501900531339628137nteger
            @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A ) @ ( if_Code_integer @ ( I3 = N ) @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) @ ( power_8256067586552552935nteger @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
thf(fact_7947_root__polyfun,axiom,
    ! [N: nat,Z: real,A: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_real @ Z @ N )
          = A )
        = ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( if_real @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I3 = N ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_real ) ) ) ).

% root_polyfun
thf(fact_7948_sum__gp0,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) )
          = ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_7949_sum__gp0,axiom,
    ! [X: rat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) )
          = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_7950_sum__gp0,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) )
          = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_7951_gbinomial__sum__nat__pow2,axiom,
    ! [M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( divide1717551699836669952omplex @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M2 @ K3 ) ) @ K3 ) @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ M2 ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_7952_gbinomial__sum__nat__pow2,axiom,
    ! [M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( divide_divide_rat @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M2 @ K3 ) ) @ K3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ M2 ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_7953_gbinomial__sum__nat__pow2,axiom,
    ! [M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( divide_divide_real @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M2 @ K3 ) ) @ K3 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ M2 ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_7954_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: complex,X: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ A ) @ K3 ) @ ( power_power_complex @ X @ K3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A ) @ one_one_complex ) @ K3 ) @ ( power_power_complex @ X @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_7955_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: rat,X: rat,Y: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ A ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A ) @ one_one_rat ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_7956_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: real,X: real,Y: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ A ) @ K3 ) @ ( power_power_real @ X @ K3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A ) @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_7957_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > complex,X: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( groups2073611262835488442omplex
                @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_complex @ Y @ K3 ) ) @ ( power_power_complex @ X @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_7958_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > rat,X: rat,Y: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_rat
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
          @ ( groups2906978787729119204at_rat
            @ ^ [J3: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_rat @ Y @ K3 ) ) @ ( power_power_rat @ X @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_7959_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > int,X: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_int @ ( minus_minus_int @ X @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( groups3539618377306564664at_int
                @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_int @ Y @ K3 ) ) @ ( power_power_int @ X @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_7960_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > real,X: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_real @ ( minus_minus_real @ X @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_real @ Y @ K3 ) ) @ ( power_power_real @ X @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_7961_polyfun__extremal__lemma,axiom,
    ! [E: real,C: nat > complex,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ? [M8: real] :
        ! [Z5: complex] :
          ( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z5 ) )
         => ( ord_less_eq_real
            @ ( real_V1022390504157884413omplex
              @ ( groups2073611262835488442omplex
                @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z5 @ I3 ) )
                @ ( set_ord_atMost_nat @ N ) ) )
            @ ( times_times_real @ E @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_7962_polyfun__extremal__lemma,axiom,
    ! [E: real,C: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ? [M8: real] :
        ! [Z5: real] :
          ( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z5 ) )
         => ( ord_less_eq_real
            @ ( real_V7735802525324610683m_real
              @ ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z5 @ I3 ) )
                @ ( set_ord_atMost_nat @ N ) ) )
            @ ( times_times_real @ E @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_7963_polyfun__diff,axiom,
    ! [N: nat,A: nat > complex,X: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
                @ ( power_power_complex @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff
thf(fact_7964_polyfun__diff,axiom,
    ! [N: nat,A: nat > rat,X: rat,Y: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_rat
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
          @ ( groups2906978787729119204at_rat
            @ ^ [J3: nat] :
                ( times_times_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
                @ ( power_power_rat @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff
thf(fact_7965_polyfun__diff,axiom,
    ! [N: nat,A: nat > int,X: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_int @ ( minus_minus_int @ X @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
                @ ( power_power_int @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff
thf(fact_7966_polyfun__diff,axiom,
    ! [N: nat,A: nat > real,X: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_real @ ( minus_minus_real @ X @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
                @ ( power_power_real @ X @ J3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff
thf(fact_7967_fact__code,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N3: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7968_fact__code,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [N3: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7969_fact__code,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N3: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7970_fact__code,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N3: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7971_gbinomial__r__part__sum,axiom,
    ! [M2: nat] :
      ( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% gbinomial_r_part_sum
thf(fact_7972_gbinomial__r__part__sum,axiom,
    ! [M2: nat] :
      ( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M2 ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% gbinomial_r_part_sum
thf(fact_7973_gbinomial__r__part__sum,axiom,
    ! [M2: nat] :
      ( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% gbinomial_r_part_sum
thf(fact_7974_choose__even__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I3 ) ) @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_even_sum
thf(fact_7975_choose__even__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) ) @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_even_sum
thf(fact_7976_choose__even__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) ) @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_even_sum
thf(fact_7977_choose__even__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I3 ) ) @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_even_sum
thf(fact_7978_choose__odd__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] :
                ( if_complex
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I3 ) )
                @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_odd_sum
thf(fact_7979_choose__odd__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] :
                ( if_int
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) )
                @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_odd_sum
thf(fact_7980_choose__odd__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] :
                ( if_rat
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) )
                @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_odd_sum
thf(fact_7981_choose__odd__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] :
                ( if_real
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I3 ) )
                @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_odd_sum
thf(fact_7982_cos__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_7983_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( sin_real @ ( arccos @ Y ) )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_7984_sin__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( sin_real @ ( arccos @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_7985_binomial__Suc__n,axiom,
    ! [N: nat] :
      ( ( binomial @ ( suc @ N ) @ N )
      = ( suc @ N ) ) ).

% binomial_Suc_n
thf(fact_7986_binomial__n__n,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ N )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_7987_binomial__0__Suc,axiom,
    ! [K2: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K2 ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_7988_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_7989_binomial__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( binomial @ N @ K2 )
        = zero_zero_nat )
      = ( ord_less_nat @ N @ K2 ) ) ).

% binomial_eq_0_iff
thf(fact_7990_binomial__Suc__Suc,axiom,
    ! [N: nat,K2: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K2 ) )
      = ( plus_plus_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_7991_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_7992_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_7993_prod__eq__1__iff,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( groups861055069439313189ex_nat @ F @ A4 )
          = one_one_nat )
        = ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
             => ( ( F @ X4 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_7994_prod__eq__1__iff,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( groups708209901874060359at_nat @ F @ A4 )
          = one_one_nat )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ( ( F @ X4 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_7995_zero__less__binomial__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K2 ) )
      = ( ord_less_eq_nat @ K2 @ N ) ) ).

% zero_less_binomial_iff
thf(fact_7996_arccos__minus__1,axiom,
    ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
    = pi ) ).

% arccos_minus_1
thf(fact_7997_prod__pos__nat__iff,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) )
        = ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_7998_prod__pos__nat__iff,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_7999_cos__arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( cos_real @ ( arccos @ Y ) )
          = Y ) ) ) ).

% cos_arccos
thf(fact_8000_sin__arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( sin_real @ ( arcsin @ Y ) )
          = Y ) ) ) ).

% sin_arcsin
thf(fact_8001_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_8002_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_8003_arcsin__minus__1,axiom,
    ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arcsin_minus_1
thf(fact_8004_choose__one,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ one_one_nat )
      = N ) ).

% choose_one
thf(fact_8005_binomial__eq__0,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ N @ K2 )
     => ( ( binomial @ N @ K2 )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_8006_Suc__times__binomial__eq,axiom,
    ! [N: nat,K2: nat] :
      ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) @ ( suc @ K2 ) ) ) ).

% Suc_times_binomial_eq
thf(fact_8007_Suc__times__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K2 ) @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) )
      = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) ) ) ).

% Suc_times_binomial
thf(fact_8008_binomial__symmetric,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( binomial @ N @ K2 )
        = ( binomial @ N @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% binomial_symmetric
thf(fact_8009_choose__mult__lemma,axiom,
    ! [M2: nat,R2: nat,K2: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R2 ) @ K2 ) @ ( plus_plus_nat @ M2 @ K2 ) ) @ ( binomial @ ( plus_plus_nat @ M2 @ K2 ) @ K2 ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R2 ) @ K2 ) @ K2 ) @ ( binomial @ ( plus_plus_nat @ M2 @ R2 ) @ M2 ) ) ) ).

% choose_mult_lemma
thf(fact_8010_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( ( ord_less_eq_nat @ R2 @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).

% binomial_le_pow
thf(fact_8011_zero__less__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K2 ) ) ) ).

% zero_less_binomial
thf(fact_8012_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_8013_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K2: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) ) @ ( suc @ K2 ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_8014_choose__mult,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( times_times_nat @ ( binomial @ N @ M2 ) @ ( binomial @ M2 @ K2 ) )
          = ( times_times_nat @ ( binomial @ N @ K2 ) @ ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M2 @ K2 ) ) ) ) ) ) ).

% choose_mult
thf(fact_8015_binomial__absorb__comp,axiom,
    ! [N: nat,K2: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N @ K2 ) @ ( binomial @ N @ K2 ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K2 ) ) ) ).

% binomial_absorb_comp
thf(fact_8016_sum__choose__upper,axiom,
    ! [M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ K3 @ M2 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ N ) @ ( suc @ M2 ) ) ) ).

% sum_choose_upper
thf(fact_8017_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_8018_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% arccos_le_mono
thf(fact_8019_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
     => ( ( ( arccos @ X )
          = ( arccos @ Y ) )
        = ( X = Y ) ) ) ).

% arccos_eq_iff
thf(fact_8020_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_8021_arcsin__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X ) )
          = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).

% arcsin_minus
thf(fact_8022_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% arcsin_le_mono
thf(fact_8023_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ( arcsin @ X )
            = ( arcsin @ Y ) )
          = ( X = Y ) ) ) ) ).

% arcsin_eq_iff
thf(fact_8024_binomial__absorption,axiom,
    ! [K2: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K2 ) ) ) ).

% binomial_absorption
thf(fact_8025_binomial__fact__lemma,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( binomial @ N @ K2 ) )
        = ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% binomial_fact_lemma
thf(fact_8026_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : X4
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_8027_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).

% sum_choose_lower
thf(fact_8028_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M2 ) @ one_one_nat ) @ M2 ) ) ).

% choose_rising_sum(2)
thf(fact_8029_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M2 ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_8030_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_8031_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_8032_binomial__mono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
       => ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K6 ) ) ) ) ).

% binomial_mono
thf(fact_8033_binomial__maximum_H,axiom,
    ! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K2 ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% binomial_maximum'
thf(fact_8034_binomial__maximum,axiom,
    ! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_8035_binomial__antimono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ K6 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K2 )
       => ( ( ord_less_eq_nat @ K6 @ N )
         => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).

% binomial_antimono
thf(fact_8036_binomial__le__pow2,axiom,
    ! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% binomial_le_pow2
thf(fact_8037_arccos__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).

% arccos_lbound
thf(fact_8038_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_8039_choose__reduce__nat,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ( binomial @ N @ K2 )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K2 ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_8040_times__binomial__minus1__eq,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( times_times_nat @ K2 @ ( binomial @ N @ K2 ) )
        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_8041_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% arccos_less_mono
thf(fact_8042_arccos__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_8043_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_8044_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% arcsin_less_mono
thf(fact_8045_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( cos_real @ ( arccos @ Y ) )
        = Y ) ) ).

% cos_arccos_abs
thf(fact_8046_binomial__altdef__nat,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( binomial @ N @ K2 )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_8047_sum__choose__diagonal,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M2 @ K3 ) )
          @ ( set_ord_atMost_nat @ M2 ) )
        = ( binomial @ ( suc @ N ) @ M2 ) ) ) ).

% sum_choose_diagonal
thf(fact_8048_vandermonde,axiom,
    ! [M2: nat,N: nat,R2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M2 @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R2 ) )
      = ( binomial @ ( plus_plus_nat @ M2 @ N ) @ R2 ) ) ).

% vandermonde
thf(fact_8049_binomial__less__binomial__Suc,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_nat @ K2 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_8050_binomial__strict__mono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( ( ord_less_nat @ K2 @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
       => ( ord_less_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K6 ) ) ) ) ).

% binomial_strict_mono
thf(fact_8051_binomial__strict__antimono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( ( ord_less_nat @ K2 @ K6 )
     => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) )
       => ( ( ord_less_eq_nat @ K6 @ N )
         => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_8052_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_8053_binomial__addition__formula,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( binomial @ N @ ( suc @ K2 ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K2 ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K2 ) ) ) ) ).

% binomial_addition_formula
thf(fact_8054_fact__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K2 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K2 ) ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ).

% fact_binomial
thf(fact_8055_fact__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ).

% fact_binomial
thf(fact_8056_fact__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K2 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ).

% fact_binomial
thf(fact_8057_binomial__fact,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K2 ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8058_binomial__fact,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8059_binomial__fact,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8060_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_real @ Y @ one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_lt_bounded
thf(fact_8061_arccos__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_8062_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ( sin_real @ ( arccos @ X ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_8063_arccos__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X ) )
          = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).

% arccos_minus
thf(fact_8064_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_8065_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_8066_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > int,N: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_8067_choose__row__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% choose_row_sum
thf(fact_8068_binomial,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial
thf(fact_8069_choose__two,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_8070_binomial__ring,axiom,
    ! [A: complex,B: complex,N: nat] :
      ( ( power_power_complex @ ( plus_plus_complex @ A @ B ) @ N )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K3 ) ) @ ( power_power_complex @ A @ K3 ) ) @ ( power_power_complex @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_8071_binomial__ring,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N )
      = ( groups3539618377306564664at_int
        @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K3 ) ) @ ( power_power_int @ A @ K3 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_8072_binomial__ring,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K3 ) ) @ ( power_power_rat @ A @ K3 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_8073_binomial__ring,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_8074_binomial__ring,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( power_power_real @ ( plus_plus_real @ A @ B ) @ N )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K3 ) ) @ ( power_power_real @ A @ K3 ) ) @ ( power_power_real @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_8075_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N )
      = ( groups3539618377306564664at_int
        @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K3 ) ) @ ( comm_s4660882817536571857er_int @ A @ K3 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8076_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8077_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K3 ) ) @ ( comm_s7457072308508201937r_real @ A @ K3 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8078_arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
          & ( ( cos_real @ ( arccos @ Y ) )
            = Y ) ) ) ) ).

% arccos
thf(fact_8079_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.triangle_reindex
thf(fact_8080_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > int,N: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J3 ) @ N ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K3 @ I3 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.triangle_reindex
thf(fact_8081_choose__square__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% choose_square_sum
thf(fact_8082_arccos__minus__abs,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X ) )
        = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).

% arccos_minus_abs
thf(fact_8083_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ I3 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
thf(fact_8084_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ I3 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
thf(fact_8085_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ I3 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_int ) ) ).

% choose_alternating_linear_sum
thf(fact_8086_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ I3 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
thf(fact_8087_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ I3 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_real ) ) ).

% choose_alternating_linear_sum
thf(fact_8088_binomial__r__part__sum,axiom,
    ! [M2: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% binomial_r_part_sum
thf(fact_8089_choose__linear__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N @ I3 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_8090_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_8091_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_real @ Y @ one_one_real )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_8092_arcsin__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).

% arcsin_lbound
thf(fact_8093_arcsin__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_8094_arcsin__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_8095_arcsin__sin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arcsin @ ( sin_real @ X ) )
          = X ) ) ) ).

% arcsin_sin
thf(fact_8096_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_complex ) ) ).

% choose_alternating_sum
thf(fact_8097_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
thf(fact_8098_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_int ) ) ).

% choose_alternating_sum
thf(fact_8099_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_rat ) ) ).

% choose_alternating_sum
thf(fact_8100_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_real ) ) ).

% choose_alternating_sum
thf(fact_8101_binomial__code,axiom,
    ( binomial
    = ( ^ [N3: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K3 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_8102_arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin
thf(fact_8103_arcsin__pi,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin_pi
thf(fact_8104_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
              = ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_8105_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
              = ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_8106_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_8107_sin__x__sin__y,axiom,
    ! [X: real,Y: real] :
      ( sums_real
      @ ^ [P5: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N3: nat] :
              ( if_real
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_real @ X @ N3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P5 @ N3 ) ) )
              @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ).

% sin_x_sin_y
thf(fact_8108_sin__x__sin__y,axiom,
    ! [X: complex,Y: complex] :
      ( sums_complex
      @ ^ [P5: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N3: nat] :
              ( if_complex
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_complex @ X @ N3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P5 @ N3 ) ) )
              @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y ) ) ) ).

% sin_x_sin_y
thf(fact_8109_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X )
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).

% Maclaurin_sin_bound
thf(fact_8110_sums__cos__x__plus__y,axiom,
    ! [X: real,Y: real] :
      ( sums_real
      @ ^ [P5: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ X @ N3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P5 @ N3 ) ) ) @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% sums_cos_x_plus_y
thf(fact_8111_sums__cos__x__plus__y,axiom,
    ! [X: complex,Y: complex] :
      ( sums_complex
      @ ^ [P5: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N3: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_complex @ X @ N3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P5 @ N3 ) ) ) @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% sums_cos_x_plus_y
thf(fact_8112_divmod__BitM__2__eq,axiom,
    ! [M2: num] :
      ( ( unique5052692396658037445od_int @ ( bitM @ M2 ) @ ( bit0 @ one ) )
      = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ one_one_int ) ) ).

% divmod_BitM_2_eq
thf(fact_8113_cos__npi__int,axiom,
    ! [N: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% cos_npi_int
thf(fact_8114_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N3: nat] : N3 ) ) ).

% of_nat_id
thf(fact_8115_mult__scaleR__right,axiom,
    ! [X: real,A: real,Y: real] :
      ( ( times_times_real @ X @ ( real_V1485227260804924795R_real @ A @ Y ) )
      = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y ) ) ) ).

% mult_scaleR_right
thf(fact_8116_mult__scaleR__right,axiom,
    ! [X: complex,A: real,Y: complex] :
      ( ( times_times_complex @ X @ ( real_V2046097035970521341omplex @ A @ Y ) )
      = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y ) ) ) ).

% mult_scaleR_right
thf(fact_8117_mult__scaleR__left,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( times_times_real @ ( real_V1485227260804924795R_real @ A @ X ) @ Y )
      = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y ) ) ) ).

% mult_scaleR_left
thf(fact_8118_mult__scaleR__left,axiom,
    ! [A: real,X: complex,Y: complex] :
      ( ( times_times_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ Y )
      = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y ) ) ) ).

% mult_scaleR_left
thf(fact_8119_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_8120_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_8121_inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) ) ) ).

% inverse_mult_distrib
thf(fact_8122_inverse__1,axiom,
    ( ( inverse_inverse_real @ one_one_real )
    = one_one_real ) ).

% inverse_1
thf(fact_8123_inverse__1,axiom,
    ( ( invers8013647133539491842omplex @ one_one_complex )
    = one_one_complex ) ).

% inverse_1
thf(fact_8124_inverse__1,axiom,
    ( ( inverse_inverse_rat @ one_one_rat )
    = one_one_rat ) ).

% inverse_1
thf(fact_8125_inverse__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( inverse_inverse_real @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% inverse_eq_1_iff
thf(fact_8126_inverse__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( ( invers8013647133539491842omplex @ X )
        = one_one_complex )
      = ( X = one_one_complex ) ) ).

% inverse_eq_1_iff
thf(fact_8127_inverse__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( inverse_inverse_rat @ X )
        = one_one_rat )
      = ( X = one_one_rat ) ) ).

% inverse_eq_1_iff
thf(fact_8128_inverse__divide,axiom,
    ! [A: real,B: real] :
      ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ B @ A ) ) ).

% inverse_divide
thf(fact_8129_inverse__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ B @ A ) ) ).

% inverse_divide
thf(fact_8130_inverse__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ B @ A ) ) ).

% inverse_divide
thf(fact_8131_scaleR__one,axiom,
    ! [X: real] :
      ( ( real_V1485227260804924795R_real @ one_one_real @ X )
      = X ) ).

% scaleR_one
thf(fact_8132_scaleR__one,axiom,
    ! [X: complex] :
      ( ( real_V2046097035970521341omplex @ one_one_real @ X )
      = X ) ).

% scaleR_one
thf(fact_8133_scaleR__scaleR,axiom,
    ! [A: real,B: real,X: real] :
      ( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B @ X ) )
      = ( real_V1485227260804924795R_real @ ( times_times_real @ A @ B ) @ X ) ) ).

% scaleR_scaleR
thf(fact_8134_scaleR__scaleR,axiom,
    ! [A: real,B: real,X: complex] :
      ( ( real_V2046097035970521341omplex @ A @ ( real_V2046097035970521341omplex @ B @ X ) )
      = ( real_V2046097035970521341omplex @ ( times_times_real @ A @ B ) @ X ) ) ).

% scaleR_scaleR
thf(fact_8135_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_8136_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% inverse_nonnegative_iff_nonnegative
thf(fact_8137_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_8138_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% inverse_nonpositive_iff_nonpositive
thf(fact_8139_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_8140_inverse__less__iff__less,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_rat @ B @ A ) ) ) ) ).

% inverse_less_iff_less
thf(fact_8141_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_8142_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_rat @ B @ A ) ) ) ) ).

% inverse_less_iff_less_neg
thf(fact_8143_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_8144_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% inverse_negative_iff_negative
thf(fact_8145_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_8146_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% inverse_positive_iff_positive
thf(fact_8147_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_8148_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_8149_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_8150_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = ( numera6690914467698888265omplex @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_8151_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_1_of_int_real @ Z )
        = ( numeral_numeral_real @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_8152_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_1_of_int_rat @ Z )
        = ( numeral_numeral_rat @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_8153_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_1_of_int_int @ Z )
        = ( numeral_numeral_int @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_8154_of__int__numeral,axiom,
    ! [K2: num] :
      ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K2 ) )
      = ( numera6690914467698888265omplex @ K2 ) ) ).

% of_int_numeral
thf(fact_8155_of__int__numeral,axiom,
    ! [K2: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_real @ K2 ) ) ).

% of_int_numeral
thf(fact_8156_of__int__numeral,axiom,
    ! [K2: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_rat @ K2 ) ) ).

% of_int_numeral
thf(fact_8157_of__int__numeral,axiom,
    ! [K2: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_int @ K2 ) ) ).

% of_int_numeral
thf(fact_8158_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_8159_scaleR__eq__iff,axiom,
    ! [B: complex,U: real,A: complex] :
      ( ( ( plus_plus_complex @ B @ ( real_V2046097035970521341omplex @ U @ A ) )
        = ( plus_plus_complex @ A @ ( real_V2046097035970521341omplex @ U @ B ) ) )
      = ( ( A = B )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
thf(fact_8160_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_8161_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_8162_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_8163_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_8164_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_8165_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_8166_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = one_one_rat )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_8167_of__int__1,axiom,
    ( ( ring_17405671764205052669omplex @ one_one_int )
    = one_one_complex ) ).

% of_int_1
thf(fact_8168_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_8169_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_8170_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_8171_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
      = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_mult
thf(fact_8172_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
      = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_mult
thf(fact_8173_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_mult
thf(fact_8174_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_add
thf(fact_8175_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_add
thf(fact_8176_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_add
thf(fact_8177_scaleR__power,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( power_power_real @ ( real_V1485227260804924795R_real @ X @ Y ) @ N )
      = ( real_V1485227260804924795R_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).

% scaleR_power
thf(fact_8178_scaleR__power,axiom,
    ! [X: real,Y: complex,N: nat] :
      ( ( power_power_complex @ ( real_V2046097035970521341omplex @ X @ Y ) @ N )
      = ( real_V2046097035970521341omplex @ ( power_power_real @ X @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).

% scaleR_power
thf(fact_8179_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_diff
thf(fact_8180_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_diff
thf(fact_8181_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_diff
thf(fact_8182_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
      = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).

% of_int_power
thf(fact_8183_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).

% of_int_power
thf(fact_8184_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).

% of_int_power
thf(fact_8185_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).

% of_int_power
thf(fact_8186_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
        = ( ring_1_of_int_rat @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_8187_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
        = ( ring_1_of_int_real @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_8188_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
        = ( ring_1_of_int_int @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_8189_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
        = ( ring_17405671764205052669omplex @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_8190_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_rat @ X )
        = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_8191_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_real @ X )
        = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_8192_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_int @ X )
        = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_8193_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_17405671764205052669omplex @ X )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_8194_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K2 ) )
      = ( numera6690914467698888265omplex @ ( bitM @ K2 ) ) ) ).

% dbl_dec_simps(5)
thf(fact_8195_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K2 ) )
      = ( numeral_numeral_real @ ( bitM @ K2 ) ) ) ).

% dbl_dec_simps(5)
thf(fact_8196_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K2 ) )
      = ( numeral_numeral_rat @ ( bitM @ K2 ) ) ) ).

% dbl_dec_simps(5)
thf(fact_8197_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_int @ ( bitM @ K2 ) ) ) ).

% dbl_dec_simps(5)
thf(fact_8198_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_8199_inverse__le__iff__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% inverse_le_iff_le
thf(fact_8200_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_8201_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% inverse_le_iff_le_neg
thf(fact_8202_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
        = one_one_real ) ) ).

% right_inverse
thf(fact_8203_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
        = one_one_complex ) ) ).

% right_inverse
thf(fact_8204_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
        = one_one_rat ) ) ).

% right_inverse
thf(fact_8205_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
        = one_one_real ) ) ).

% left_inverse
thf(fact_8206_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
        = one_one_complex ) ) ).

% left_inverse
thf(fact_8207_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
        = one_one_rat ) ) ).

% left_inverse
thf(fact_8208_inverse__eq__divide__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_real @ ( numeral_numeral_real @ W ) )
      = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_8209_inverse__eq__divide__numeral,axiom,
    ! [W: num] :
      ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_8210_inverse__eq__divide__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_8211_scaleR__minus1__left,axiom,
    ! [X: real] :
      ( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ one_one_real ) @ X )
      = ( uminus_uminus_real @ X ) ) ).

% scaleR_minus1_left
thf(fact_8212_scaleR__minus1__left,axiom,
    ! [X: complex] :
      ( ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ one_one_real ) @ X )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% scaleR_minus1_left
thf(fact_8213_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_8214_scaleR__collapse,axiom,
    ! [U: real,A: complex] :
      ( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V2046097035970521341omplex @ U @ A ) )
      = A ) ).

% scaleR_collapse
thf(fact_8215_norm__scaleR,axiom,
    ! [A: real,X: real] :
      ( ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ A @ X ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V7735802525324610683m_real @ X ) ) ) ).

% norm_scaleR
thf(fact_8216_norm__scaleR,axiom,
    ! [A: real,X: complex] :
      ( ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V1022390504157884413omplex @ X ) ) ) ).

% norm_scaleR
thf(fact_8217_pred__numeral__simps_I2_J,axiom,
    ! [K2: num] :
      ( ( pred_numeral @ ( bit0 @ K2 ) )
      = ( numeral_numeral_nat @ ( bitM @ K2 ) ) ) ).

% pred_numeral_simps(2)
thf(fact_8218_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_8219_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] :
      ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_8220_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_8221_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_8222_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_8223_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_8224_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_8225_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_8226_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_8227_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_8228_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_8229_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_8230_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_8231_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_8232_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_8233_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_8234_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_8235_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_8236_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_8237_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_8238_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_8239_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_8240_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_8241_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_8242_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_8243_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_8244_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_8245_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_8246_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_8247_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_8248_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_8249_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_8250_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_8251_scaleR__times,axiom,
    ! [U: num,W: num,A: real] :
      ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ U ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( real_V1485227260804924795R_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).

% scaleR_times
thf(fact_8252_scaleR__times,axiom,
    ! [U: num,W: num,A: complex] :
      ( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ U ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( real_V2046097035970521341omplex @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).

% scaleR_times
thf(fact_8253_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_8254_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_8255_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_8256_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_8257_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_8258_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_8259_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_8260_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_8261_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_8262_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_8263_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_8264_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_8265_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_8266_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_8267_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_8268_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_8269_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_8270_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_8271_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_8272_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_8273_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_8274_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_8275_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_8276_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_8277_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_8278_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_8279_sin__npi__int,axiom,
    ! [N: int] :
      ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi_int
thf(fact_8280_tan__periodic__int,axiom,
    ! [X: real,I: int] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_int
thf(fact_8281_inverse__scaleR__times,axiom,
    ! [V: num,W: num,A: real] :
      ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).

% inverse_scaleR_times
thf(fact_8282_inverse__scaleR__times,axiom,
    ! [V: num,W: num,A: complex] :
      ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).

% inverse_scaleR_times
thf(fact_8283_fraction__scaleR__times,axiom,
    ! [U: num,V: num,W: num,A: real] :
      ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).

% fraction_scaleR_times
thf(fact_8284_fraction__scaleR__times,axiom,
    ! [U: num,V: num,W: num,A: complex] :
      ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).

% fraction_scaleR_times
thf(fact_8285_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_8286_scaleR__half__double,axiom,
    ! [A: complex] :
      ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ A @ A ) )
      = A ) ).

% scaleR_half_double
thf(fact_8287_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_8288_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_8289_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_8290_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_8291_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_8292_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_8293_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_8294_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_8295_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_8296_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_8297_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_8298_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_8299_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_8300_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_8301_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_8302_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_8303_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_8304_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_8305_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_8306_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_8307_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_8308_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_8309_sin__int__2pin,axiom,
    ! [N: int] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
      = zero_zero_real ) ).

% sin_int_2pin
thf(fact_8310_cos__int__2pin,axiom,
    ! [N: int] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
      = one_one_real ) ).

% cos_int_2pin
thf(fact_8311_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_8312_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_8313_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_8314_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_8315_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_8316_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_8317_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_8318_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_8319_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_8320_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_8321_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_8322_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_8323_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_8324_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_8325_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_8326_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_8327_real__scaleR__def,axiom,
    real_V1485227260804924795R_real = times_times_real ).

% real_scaleR_def
thf(fact_8328_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_8329_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] :
      ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).

% mult_of_int_commute
thf(fact_8330_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_8331_power__inverse,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( inverse_inverse_real @ A ) @ N )
      = ( inverse_inverse_real @ ( power_power_real @ A @ N ) ) ) ).

% power_inverse
thf(fact_8332_power__inverse,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( invers8013647133539491842omplex @ A ) @ N )
      = ( invers8013647133539491842omplex @ ( power_power_complex @ A @ N ) ) ) ).

% power_inverse
thf(fact_8333_power__inverse,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( inverse_inverse_rat @ A ) @ N )
      = ( inverse_inverse_rat @ ( power_power_rat @ A @ N ) ) ) ).

% power_inverse
thf(fact_8334_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z3: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).

% ex_less_of_int
thf(fact_8335_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z3: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).

% ex_less_of_int
thf(fact_8336_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ).

% ex_of_int_less
thf(fact_8337_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z3: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ X ) ).

% ex_of_int_less
thf(fact_8338_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z3: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).

% ex_le_of_int
thf(fact_8339_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z3: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).

% ex_le_of_int
thf(fact_8340_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_8341_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_8342_mult__inverse__of__int__commute,axiom,
    ! [Xa2: int,X: rat] :
      ( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa2 ) ) @ X )
      = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa2 ) ) ) ) ).

% mult_inverse_of_int_commute
thf(fact_8343_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_8344_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_8345_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: rat,X: rat] :
      ( ( ( times_times_rat @ Y @ X )
        = ( times_times_rat @ X @ Y ) )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ Y ) @ X )
        = ( times_times_rat @ X @ ( inverse_inverse_rat @ Y ) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
thf(fact_8346_scaleR__right__diff__distrib,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( real_V1485227260804924795R_real @ A @ ( minus_minus_real @ X @ Y ) )
      = ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).

% scaleR_right_diff_distrib
thf(fact_8347_scaleR__right__diff__distrib,axiom,
    ! [A: real,X: complex,Y: complex] :
      ( ( real_V2046097035970521341omplex @ A @ ( minus_minus_complex @ X @ Y ) )
      = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ A @ Y ) ) ) ).

% scaleR_right_diff_distrib
thf(fact_8348_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_8349_scaleR__right__distrib,axiom,
    ! [A: real,X: complex,Y: complex] :
      ( ( real_V2046097035970521341omplex @ A @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ A @ Y ) ) ) ).

% scaleR_right_distrib
thf(fact_8350_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_8351_real__vector__affinity__eq,axiom,
    ! [M2: real,X: real,C: real,Y: real] :
      ( ( M2 != zero_zero_real )
     => ( ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ M2 @ X ) @ C )
          = Y )
        = ( X
          = ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M2 ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M2 ) @ C ) ) ) ) ) ).

% real_vector_affinity_eq
thf(fact_8352_real__vector__affinity__eq,axiom,
    ! [M2: real,X: complex,C: complex,Y: complex] :
      ( ( M2 != zero_zero_real )
     => ( ( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ M2 @ X ) @ C )
          = Y )
        = ( X
          = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M2 ) @ Y ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M2 ) @ C ) ) ) ) ) ).

% real_vector_affinity_eq
thf(fact_8353_real__vector__eq__affinity,axiom,
    ! [M2: real,Y: real,X: real,C: real] :
      ( ( M2 != zero_zero_real )
     => ( ( Y
          = ( plus_plus_real @ ( real_V1485227260804924795R_real @ M2 @ X ) @ C ) )
        = ( ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M2 ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M2 ) @ C ) )
          = X ) ) ) ).

% real_vector_eq_affinity
thf(fact_8354_real__vector__eq__affinity,axiom,
    ! [M2: real,Y: complex,X: complex,C: complex] :
      ( ( M2 != zero_zero_real )
     => ( ( Y
          = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ M2 @ X ) @ C ) )
        = ( ( minus_minus_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M2 ) @ Y ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M2 ) @ C ) )
          = X ) ) ) ).

% real_vector_eq_affinity
thf(fact_8355_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_8356_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_8357_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_8358_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_8359_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_8360_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_8361_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_8362_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_8363_summable__exp__generic,axiom,
    ! [X: real] :
      ( summable_real
      @ ^ [N3: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ).

% summable_exp_generic
thf(fact_8364_summable__exp__generic,axiom,
    ! [X: complex] :
      ( summable_complex
      @ ^ [N3: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_complex @ X @ N3 ) ) ) ).

% summable_exp_generic
thf(fact_8365_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_8366_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_8367_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_8368_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_8369_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_8370_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_8371_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_8372_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_8373_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_8374_inverse__less__imp__less,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ B @ A ) ) ) ).

% inverse_less_imp_less
thf(fact_8375_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_8376_less__imp__inverse__less,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% less_imp_inverse_less
thf(fact_8377_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_8378_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ B @ A ) ) ) ).

% inverse_less_imp_less_neg
thf(fact_8379_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_8380_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% less_imp_inverse_less_neg
thf(fact_8381_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_8382_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
     => ( ( A != zero_zero_rat )
       => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% inverse_negative_imp_negative
thf(fact_8383_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_8384_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
     => ( ( A != zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).

% inverse_positive_imp_positive
thf(fact_8385_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_8386_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).

% negative_imp_inverse_negative
thf(fact_8387_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_8388_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).

% positive_imp_inverse_positive
thf(fact_8389_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_8390_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_8391_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
          = ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_8392_inverse__numeral__1,axiom,
    ( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
    = ( numeral_numeral_real @ one ) ) ).

% inverse_numeral_1
thf(fact_8393_inverse__numeral__1,axiom,
    ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( numera6690914467698888265omplex @ one ) ) ).

% inverse_numeral_1
thf(fact_8394_inverse__numeral__1,axiom,
    ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
    = ( numeral_numeral_rat @ one ) ) ).

% inverse_numeral_1
thf(fact_8395_inverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = one_one_real )
     => ( ( inverse_inverse_real @ A )
        = B ) ) ).

% inverse_unique
thf(fact_8396_inverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = one_one_complex )
     => ( ( invers8013647133539491842omplex @ A )
        = B ) ) ).

% inverse_unique
thf(fact_8397_inverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = one_one_rat )
     => ( ( inverse_inverse_rat @ A )
        = B ) ) ).

% inverse_unique
thf(fact_8398_divide__inverse__commute,axiom,
    ( divide_divide_real
    = ( ^ [A5: real,B5: real] : ( times_times_real @ ( inverse_inverse_real @ B5 ) @ A5 ) ) ) ).

% divide_inverse_commute
thf(fact_8399_divide__inverse__commute,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A5: complex,B5: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B5 ) @ A5 ) ) ) ).

% divide_inverse_commute
thf(fact_8400_divide__inverse__commute,axiom,
    ( divide_divide_rat
    = ( ^ [A5: rat,B5: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B5 ) @ A5 ) ) ) ).

% divide_inverse_commute
thf(fact_8401_divide__inverse,axiom,
    ( divide_divide_real
    = ( ^ [A5: real,B5: real] : ( times_times_real @ A5 @ ( inverse_inverse_real @ B5 ) ) ) ) ).

% divide_inverse
thf(fact_8402_divide__inverse,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A5: complex,B5: complex] : ( times_times_complex @ A5 @ ( invers8013647133539491842omplex @ B5 ) ) ) ) ).

% divide_inverse
thf(fact_8403_divide__inverse,axiom,
    ( divide_divide_rat
    = ( ^ [A5: rat,B5: rat] : ( times_times_rat @ A5 @ ( inverse_inverse_rat @ B5 ) ) ) ) ).

% divide_inverse
thf(fact_8404_field__class_Ofield__divide__inverse,axiom,
    ( divide_divide_real
    = ( ^ [A5: real,B5: real] : ( times_times_real @ A5 @ ( inverse_inverse_real @ B5 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_8405_field__class_Ofield__divide__inverse,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A5: complex,B5: complex] : ( times_times_complex @ A5 @ ( invers8013647133539491842omplex @ B5 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_8406_field__class_Ofield__divide__inverse,axiom,
    ( divide_divide_rat
    = ( ^ [A5: rat,B5: rat] : ( times_times_rat @ A5 @ ( inverse_inverse_rat @ B5 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_8407_inverse__eq__divide,axiom,
    ( inverse_inverse_real
    = ( divide_divide_real @ one_one_real ) ) ).

% inverse_eq_divide
thf(fact_8408_inverse__eq__divide,axiom,
    ( invers8013647133539491842omplex
    = ( divide1717551699836669952omplex @ one_one_complex ) ) ).

% inverse_eq_divide
thf(fact_8409_inverse__eq__divide,axiom,
    ( inverse_inverse_rat
    = ( divide_divide_rat @ one_one_rat ) ) ).

% inverse_eq_divide
thf(fact_8410_power__mult__power__inverse__commute,axiom,
    ! [X: real,M2: nat,N: nat] :
      ( ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) )
      = ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) @ ( power_power_real @ X @ M2 ) ) ) ).

% power_mult_power_inverse_commute
thf(fact_8411_power__mult__power__inverse__commute,axiom,
    ! [X: complex,M2: nat,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) )
      = ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) @ ( power_power_complex @ X @ M2 ) ) ) ).

% power_mult_power_inverse_commute
thf(fact_8412_power__mult__power__inverse__commute,axiom,
    ! [X: rat,M2: nat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) )
      = ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) @ ( power_power_rat @ X @ M2 ) ) ) ).

% power_mult_power_inverse_commute
thf(fact_8413_power__mult__inverse__distrib,axiom,
    ! [X: real,M2: nat] :
      ( ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( inverse_inverse_real @ X ) )
      = ( times_times_real @ ( inverse_inverse_real @ X ) @ ( power_power_real @ X @ M2 ) ) ) ).

% power_mult_inverse_distrib
thf(fact_8414_power__mult__inverse__distrib,axiom,
    ! [X: complex,M2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( invers8013647133539491842omplex @ X ) )
      = ( times_times_complex @ ( invers8013647133539491842omplex @ X ) @ ( power_power_complex @ X @ M2 ) ) ) ).

% power_mult_inverse_distrib
thf(fact_8415_power__mult__inverse__distrib,axiom,
    ! [X: rat,M2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( inverse_inverse_rat @ X ) )
      = ( times_times_rat @ ( inverse_inverse_rat @ X ) @ ( power_power_rat @ X @ M2 ) ) ) ).

% power_mult_inverse_distrib
thf(fact_8416_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_8417_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_8418_mult__inverse__of__nat__commute,axiom,
    ! [Xa2: nat,X: rat] :
      ( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa2 ) ) @ X )
      = ( times_times_rat @ X @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa2 ) ) ) ) ).

% mult_inverse_of_nat_commute
thf(fact_8419_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_8420_scaleR__left__distrib,axiom,
    ! [A: real,B: real,X: complex] :
      ( ( real_V2046097035970521341omplex @ ( plus_plus_real @ A @ B ) @ X )
      = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ B @ X ) ) ) ).

% scaleR_left_distrib
thf(fact_8421_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_8422_scaleR__left_Oadd,axiom,
    ! [X: real,Y: real,Xa2: complex] :
      ( ( real_V2046097035970521341omplex @ ( plus_plus_real @ X @ Y ) @ Xa2 )
      = ( plus_plus_complex @ ( real_V2046097035970521341omplex @ X @ Xa2 ) @ ( real_V2046097035970521341omplex @ Y @ Xa2 ) ) ) ).

% scaleR_left.add
thf(fact_8423_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X4: real,Y3: real] : ( times_times_real @ X4 @ ( inverse_inverse_real @ Y3 ) ) ) ) ).

% divide_real_def
thf(fact_8424_scaleR__left__diff__distrib,axiom,
    ! [A: real,B: real,X: real] :
      ( ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ X )
      = ( minus_minus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ).

% scaleR_left_diff_distrib
thf(fact_8425_scaleR__left__diff__distrib,axiom,
    ! [A: real,B: real,X: complex] :
      ( ( real_V2046097035970521341omplex @ ( minus_minus_real @ A @ B ) @ X )
      = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ B @ X ) ) ) ).

% scaleR_left_diff_distrib
thf(fact_8426_scaleR__left_Odiff,axiom,
    ! [X: real,Y: real,Xa2: real] :
      ( ( real_V1485227260804924795R_real @ ( minus_minus_real @ X @ Y ) @ Xa2 )
      = ( minus_minus_real @ ( real_V1485227260804924795R_real @ X @ Xa2 ) @ ( real_V1485227260804924795R_real @ Y @ Xa2 ) ) ) ).

% scaleR_left.diff
thf(fact_8427_scaleR__left_Odiff,axiom,
    ! [X: real,Y: real,Xa2: complex] :
      ( ( real_V2046097035970521341omplex @ ( minus_minus_real @ X @ Y ) @ Xa2 )
      = ( minus_minus_complex @ ( real_V2046097035970521341omplex @ X @ Xa2 ) @ ( real_V2046097035970521341omplex @ Y @ Xa2 ) ) ) ).

% scaleR_left.diff
thf(fact_8428_exp__converges,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) )
      @ ( exp_real @ X ) ) ).

% exp_converges
thf(fact_8429_exp__converges,axiom,
    ! [X: complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_complex @ X @ N3 ) )
      @ ( exp_complex @ X ) ) ).

% exp_converges
thf(fact_8430_exp__def,axiom,
    ( exp_real
    = ( ^ [X4: real] :
          ( suminf_real
          @ ^ [N3: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X4 @ N3 ) ) ) ) ) ).

% exp_def
thf(fact_8431_exp__def,axiom,
    ( exp_complex
    = ( ^ [X4: complex] :
          ( suminf_complex
          @ ^ [N3: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_complex @ X4 @ N3 ) ) ) ) ) ).

% exp_def
thf(fact_8432_summable__norm__exp,axiom,
    ! [X: real] :
      ( summable_real
      @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ).

% summable_norm_exp
thf(fact_8433_summable__norm__exp,axiom,
    ! [X: complex] :
      ( summable_real
      @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_complex @ X @ N3 ) ) ) ) ).

% summable_norm_exp
thf(fact_8434_complex__scaleR,axiom,
    ! [R2: real,A: real,B: real] :
      ( ( real_V2046097035970521341omplex @ R2 @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ B ) ) ) ).

% complex_scaleR
thf(fact_8435_semiring__norm_I27_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit0 @ N ) )
      = ( bit1 @ ( bitM @ N ) ) ) ).

% semiring_norm(27)
thf(fact_8436_semiring__norm_I28_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ N ) ) ) ).

% semiring_norm(28)
thf(fact_8437_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_8438_inverse__le__imp__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ B @ A ) ) ) ).

% inverse_le_imp_le
thf(fact_8439_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_8440_le__imp__inverse__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% le_imp_inverse_le
thf(fact_8441_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_8442_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ B @ A ) ) ) ).

% inverse_le_imp_le_neg
thf(fact_8443_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_8444_le__imp__inverse__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% le_imp_inverse_le_neg
thf(fact_8445_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_8446_inverse__le__1__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
      = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% inverse_le_1_iff
thf(fact_8447_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_8448_one__less__inverse,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).

% one_less_inverse
thf(fact_8449_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_8450_one__less__inverse__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
      = ( ( ord_less_rat @ zero_zero_rat @ X )
        & ( ord_less_rat @ X @ one_one_rat ) ) ) ).

% one_less_inverse_iff
thf(fact_8451_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_8452_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_8453_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
        = one_one_rat ) ) ).

% field_class.field_inverse
thf(fact_8454_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_8455_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_8456_division__ring__inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_8457_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_8458_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_8459_inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).

% inverse_add
thf(fact_8460_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_8461_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_8462_division__ring__inverse__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).

% division_ring_inverse_diff
thf(fact_8463_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_8464_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ A )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_8465_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ A )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_8466_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_8467_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_8468_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_8469_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_8470_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_8471_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_8472_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_8473_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% Real_Vector_Spaces.le_add_iff2
thf(fact_8474_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).

% Real_Vector_Spaces.le_add_iff1
thf(fact_8475_real__of__int__div4,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).

% real_of_int_div4
thf(fact_8476_real__of__int__div,axiom,
    ! [D: int,N: int] :
      ( ( dvd_dvd_int @ D @ N )
     => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
        = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div
thf(fact_8477_eval__nat__numeral_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).

% eval_nat_numeral(2)
thf(fact_8478_exp__series__add__commuting,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ N ) )
        = ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ I3 ) ) @ ( power_power_real @ X @ I3 ) ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ I3 ) ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ I3 ) ) ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% exp_series_add_commuting
thf(fact_8479_exp__series__add__commuting,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ N ) )
        = ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ I3 ) ) @ ( power_power_complex @ X @ I3 ) ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ I3 ) ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ I3 ) ) ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% exp_series_add_commuting
thf(fact_8480_exp__first__term,axiom,
    ( exp_real
    = ( ^ [X4: real] :
          ( plus_plus_real @ one_one_real
          @ ( suminf_real
            @ ^ [N3: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X4 @ ( suc @ N3 ) ) ) ) ) ) ) ).

% exp_first_term
thf(fact_8481_exp__first__term,axiom,
    ( exp_complex
    = ( ^ [X4: complex] :
          ( plus_plus_complex @ one_one_complex
          @ ( suminf_complex
            @ ^ [N3: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( suc @ N3 ) ) ) @ ( power_power_complex @ X4 @ ( suc @ N3 ) ) ) ) ) ) ) ).

% exp_first_term
thf(fact_8482_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_8483_inverse__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ B @ A ) )
        & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
         => ( ord_less_eq_rat @ A @ B ) ) ) ) ).

% inverse_le_iff
thf(fact_8484_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_8485_inverse__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ B @ A ) )
        & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
         => ( ord_less_rat @ A @ B ) ) ) ) ).

% inverse_less_iff
thf(fact_8486_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_8487_one__le__inverse,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).

% one_le_inverse
thf(fact_8488_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_8489_inverse__less__1__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
      = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% inverse_less_1_iff
thf(fact_8490_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_8491_one__le__inverse__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
      = ( ( ord_less_rat @ zero_zero_rat @ X )
        & ( ord_less_eq_rat @ X @ one_one_rat ) ) ) ).

% one_le_inverse_iff
thf(fact_8492_one__plus__BitM,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% one_plus_BitM
thf(fact_8493_BitM__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ ( bitM @ N ) @ one )
      = ( bit0 @ N ) ) ).

% BitM_plus_one
thf(fact_8494_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_8495_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_8496_inverse__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ) ).

% inverse_diff_inverse
thf(fact_8497_reals__Archimedean,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N2: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ X ) ) ).

% reals_Archimedean
thf(fact_8498_reals__Archimedean,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N2: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ X ) ) ).

% reals_Archimedean
thf(fact_8499_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_8500_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_nonneg
thf(fact_8501_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_8502_of__int__leD,axiom,
    ! [N: int,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_leD
thf(fact_8503_of__int__leD,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% of_int_leD
thf(fact_8504_of__int__leD,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% of_int_leD
thf(fact_8505_of__int__leD,axiom,
    ! [N: int,X: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).

% of_int_leD
thf(fact_8506_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_8507_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_pos
thf(fact_8508_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_8509_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_8510_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_8511_of__int__lessD,axiom,
    ! [N: int,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_lessD
thf(fact_8512_of__int__lessD,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X ) ) ) ).

% of_int_lessD
thf(fact_8513_of__int__lessD,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% of_int_lessD
thf(fact_8514_of__int__lessD,axiom,
    ! [N: int,X: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X ) ) ) ).

% of_int_lessD
thf(fact_8515_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_8516_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_8517_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_8518_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_8519_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_8520_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_8521_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_8522_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_8523_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_8524_floor__exists,axiom,
    ! [X: real] :
    ? [Z3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_8525_floor__exists,axiom,
    ! [X: rat] :
    ? [Z3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_8526_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 ) ) )
      & ! [Y5: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X )
            & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
         => ( Y5 = X5 ) ) ) ).

% floor_exists1
thf(fact_8527_floor__exists1,axiom,
    ! [X: rat] :
    ? [X5: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X5 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X5 @ one_one_int ) ) )
      & ! [Y5: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
         => ( Y5 = X5 ) ) ) ).

% floor_exists1
thf(fact_8528_of__int__neg__numeral,axiom,
    ! [K2: num] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) ) ).

% of_int_neg_numeral
thf(fact_8529_of__int__neg__numeral,axiom,
    ! [K2: num] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) ).

% of_int_neg_numeral
thf(fact_8530_of__int__neg__numeral,axiom,
    ! [K2: num] :
      ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K2 ) ) ) ).

% of_int_neg_numeral
thf(fact_8531_of__int__neg__numeral,axiom,
    ! [K2: num] :
      ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) ) ) ).

% of_int_neg_numeral
thf(fact_8532_of__int__neg__numeral,axiom,
    ! [K2: num] :
      ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K2 ) ) ) ).

% of_int_neg_numeral
thf(fact_8533_scaleR__2,axiom,
    ! [X: real] :
      ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
      = ( plus_plus_real @ X @ X ) ) ).

% scaleR_2
thf(fact_8534_scaleR__2,axiom,
    ! [X: complex] :
      ( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
      = ( plus_plus_complex @ X @ X ) ) ).

% scaleR_2
thf(fact_8535_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_8536_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_8537_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_8538_forall__pos__mono__1,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono_1
thf(fact_8539_real__arch__inverse,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
      = ( ? [N3: nat] :
            ( ( N3 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E ) ) ) ) ).

% real_arch_inverse
thf(fact_8540_forall__pos__mono,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono
thf(fact_8541_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N3: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_8542_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N3: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).

% int_less_real_le
thf(fact_8543_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( X
            = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_8544_exp__first__terms,axiom,
    ! [K2: nat] :
      ( exp_real
      = ( ^ [X4: real] :
            ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [N3: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X4 @ N3 ) )
              @ ( set_ord_lessThan_nat @ K2 ) )
            @ ( suminf_real
              @ ^ [N3: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N3 @ K2 ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ N3 @ K2 ) ) ) ) ) ) ) ).

% exp_first_terms
thf(fact_8545_exp__first__terms,axiom,
    ! [K2: nat] :
      ( exp_complex
      = ( ^ [X4: complex] :
            ( plus_plus_complex
            @ ( groups2073611262835488442omplex
              @ ^ [N3: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_complex @ X4 @ N3 ) )
              @ ( set_ord_lessThan_nat @ K2 ) )
            @ ( suminf_complex
              @ ^ [N3: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N3 @ K2 ) ) ) @ ( power_power_complex @ X4 @ ( plus_plus_nat @ N3 @ K2 ) ) ) ) ) ) ) ).

% exp_first_terms
thf(fact_8546_real__of__int__div__aux,axiom,
    ! [X: int,D: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div_aux
thf(fact_8547_summable__exp,axiom,
    ! [X: complex] :
      ( summable_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri5044797733671781792omplex @ N3 ) ) @ ( power_power_complex @ X @ N3 ) ) ) ).

% summable_exp
thf(fact_8548_summable__exp,axiom,
    ! [X: real] :
      ( summable_real
      @ ^ [N3: nat] : ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ).

% summable_exp
thf(fact_8549_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bitM @ N ) )
      = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).

% numeral_BitM
thf(fact_8550_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bitM @ N ) )
      = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).

% numeral_BitM
thf(fact_8551_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bitM @ N ) )
      = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).

% numeral_BitM
thf(fact_8552_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bitM @ N ) )
      = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).

% numeral_BitM
thf(fact_8553_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_8554_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_8555_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_8556_ex__inverse__of__nat__less,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X ) ) ) ).

% ex_inverse_of_nat_less
thf(fact_8557_ex__inverse__of__nat__less,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ X ) ) ) ).

% ex_inverse_of_nat_less
thf(fact_8558_power__diff__conv__inverse,axiom,
    ! [X: real,M2: nat,N: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( power_power_real @ X @ ( minus_minus_nat @ N @ M2 ) )
          = ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M2 ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_8559_power__diff__conv__inverse,axiom,
    ! [X: complex,M2: nat,N: nat] :
      ( ( X != zero_zero_complex )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( power_power_complex @ X @ ( minus_minus_nat @ N @ M2 ) )
          = ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M2 ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_8560_power__diff__conv__inverse,axiom,
    ! [X: rat,M2: nat,N: nat] :
      ( ( X != zero_zero_rat )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( power_power_rat @ X @ ( minus_minus_nat @ N @ M2 ) )
          = ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M2 ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_8561_sin__converges,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( real_V1485227260804924795R_real @ ( sin_coeff @ N3 ) @ ( power_power_real @ X @ N3 ) )
      @ ( sin_real @ X ) ) ).

% sin_converges
thf(fact_8562_sin__converges,axiom,
    ! [X: complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( real_V2046097035970521341omplex @ ( sin_coeff @ N3 ) @ ( power_power_complex @ X @ N3 ) )
      @ ( sin_complex @ X ) ) ).

% sin_converges
thf(fact_8563_sin__def,axiom,
    ( sin_real
    = ( ^ [X4: real] :
          ( suminf_real
          @ ^ [N3: nat] : ( real_V1485227260804924795R_real @ ( sin_coeff @ N3 ) @ ( power_power_real @ X4 @ N3 ) ) ) ) ) ).

% sin_def
thf(fact_8564_sin__def,axiom,
    ( sin_complex
    = ( ^ [X4: complex] :
          ( suminf_complex
          @ ^ [N3: nat] : ( real_V2046097035970521341omplex @ ( sin_coeff @ N3 ) @ ( power_power_complex @ X4 @ N3 ) ) ) ) ) ).

% sin_def
thf(fact_8565_cos__converges,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N3 ) @ ( power_power_real @ X @ N3 ) )
      @ ( cos_real @ X ) ) ).

% cos_converges
thf(fact_8566_cos__converges,axiom,
    ! [X: complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N3 ) @ ( power_power_complex @ X @ N3 ) )
      @ ( cos_complex @ X ) ) ).

% cos_converges
thf(fact_8567_cos__def,axiom,
    ( cos_real
    = ( ^ [X4: real] :
          ( suminf_real
          @ ^ [N3: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N3 ) @ ( power_power_real @ X4 @ N3 ) ) ) ) ) ).

% cos_def
thf(fact_8568_cos__def,axiom,
    ( cos_complex
    = ( ^ [X4: complex] :
          ( suminf_complex
          @ ^ [N3: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N3 ) @ ( power_power_complex @ X4 @ N3 ) ) ) ) ) ).

% cos_def
thf(fact_8569_summable__norm__sin,axiom,
    ! [X: real] :
      ( summable_real
      @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( sin_coeff @ N3 ) @ ( power_power_real @ X @ N3 ) ) ) ) ).

% summable_norm_sin
thf(fact_8570_summable__norm__sin,axiom,
    ! [X: complex] :
      ( summable_real
      @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( sin_coeff @ N3 ) @ ( power_power_complex @ X @ N3 ) ) ) ) ).

% summable_norm_sin
thf(fact_8571_summable__norm__cos,axiom,
    ! [X: real] :
      ( summable_real
      @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( cos_coeff @ N3 ) @ ( power_power_real @ X @ N3 ) ) ) ) ).

% summable_norm_cos
thf(fact_8572_summable__norm__cos,axiom,
    ! [X: complex] :
      ( summable_real
      @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( cos_coeff @ N3 ) @ ( power_power_complex @ X @ N3 ) ) ) ) ).

% summable_norm_cos
thf(fact_8573_exp__first__two__terms,axiom,
    ( exp_real
    = ( ^ [X4: real] :
          ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X4 )
          @ ( suminf_real
            @ ^ [N3: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% exp_first_two_terms
thf(fact_8574_exp__first__two__terms,axiom,
    ( exp_complex
    = ( ^ [X4: complex] :
          ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ X4 )
          @ ( suminf_complex
            @ ^ [N3: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_complex @ X4 @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% exp_first_two_terms
thf(fact_8575_real__of__int__div2,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).

% real_of_int_div2
thf(fact_8576_real__of__int__div3,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_8577_sin__minus__converges,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( sin_coeff @ N3 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ N3 ) ) )
      @ ( sin_real @ X ) ) ).

% sin_minus_converges
thf(fact_8578_sin__minus__converges,axiom,
    ! [X: complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ ( sin_coeff @ N3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ N3 ) ) )
      @ ( sin_complex @ X ) ) ).

% sin_minus_converges
thf(fact_8579_cos__minus__converges,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N3 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ N3 ) )
      @ ( cos_real @ X ) ) ).

% cos_minus_converges
thf(fact_8580_cos__minus__converges,axiom,
    ! [X: complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ N3 ) )
      @ ( cos_complex @ X ) ) ).

% cos_minus_converges
thf(fact_8581_even__of__int__iff,axiom,
    ! [K2: int] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) ).

% even_of_int_iff
thf(fact_8582_even__of__int__iff,axiom,
    ! [K2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) ).

% even_of_int_iff
thf(fact_8583_exp__plus__inverse__exp,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_8584_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_8585_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_8586_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X4: int] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_8587_tan__cot,axiom,
    ! [X: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
      = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).

% tan_cot
thf(fact_8588_real__le__x__sinh,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_8589_real__le__abs__sinh,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_8590_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_8591_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_8592_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K: int] :
          ( ( arccos @ ( cos_real @ Theta ) )
         != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).

% arccos_cos_eq_abs_2pi
thf(fact_8593_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_8594_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_8595_cos__x__cos__y,axiom,
    ! [X: real,Y: real] :
      ( sums_real
      @ ^ [P5: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N3: nat] :
              ( if_real
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ X @ N3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P5 @ N3 ) ) )
              @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ).

% cos_x_cos_y
thf(fact_8596_cos__x__cos__y,axiom,
    ! [X: complex,Y: complex] :
      ( sums_complex
      @ ^ [P5: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N3: nat] :
              ( if_complex
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N3 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_complex @ X @ N3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P5 @ N3 ) ) )
              @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ).

% cos_x_cos_y
thf(fact_8597_of__int__code__if,axiom,
    ( ring_1_of_int_real
    = ( ^ [K3: int] :
          ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
          @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_real
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_8598_of__int__code__if,axiom,
    ( ring_1_of_int_int
    = ( ^ [K3: int] :
          ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
          @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_int
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_8599_of__int__code__if,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K3: int] :
          ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
          @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_complex
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_8600_of__int__code__if,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K3: int] :
          ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
          @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_rat
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_8601_of__int__code__if,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
          @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_8602_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_8603_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_8604_round__unique_H,axiom,
    ! [X: real,N: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X )
        = N ) ) ).

% round_unique'
thf(fact_8605_round__unique_H,axiom,
    ! [X: rat,N: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X )
        = N ) ) ).

% round_unique'
thf(fact_8606_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_8607_of__int__round__abs__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_8608_sinh__converges,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) )
      @ ( sinh_real @ X ) ) ).

% sinh_converges
thf(fact_8609_sinh__converges,axiom,
    ! [X: complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_complex @ X @ N3 ) ) )
      @ ( sinh_complex @ X ) ) ).

% sinh_converges
thf(fact_8610_round__numeral,axiom,
    ! [N: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_8611_round__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_8612_round__1,axiom,
    ( ( archim8280529875227126926d_real @ one_one_real )
    = one_one_int ) ).

% round_1
thf(fact_8613_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_8614_round__neg__numeral,axiom,
    ! [N: num] :
      ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% round_neg_numeral
thf(fact_8615_round__neg__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% round_neg_numeral
thf(fact_8616_divide__complex__def,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X4: complex,Y3: complex] : ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ Y3 ) ) ) ) ).

% divide_complex_def
thf(fact_8617_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).

% round_mono
thf(fact_8618_round__diff__minimal,axiom,
    ! [Z: real,M2: 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 @ M2 ) ) ) ) ).

% round_diff_minimal
thf(fact_8619_round__diff__minimal,axiom,
    ! [Z: rat,M2: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M2 ) ) ) ) ).

% round_diff_minimal
thf(fact_8620_complex__inverse,axiom,
    ! [A: real,B: real] :
      ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_inverse
thf(fact_8621_sinh__field__def,axiom,
    ( sinh_real
    = ( ^ [Z2: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z2 ) @ ( exp_real @ ( uminus_uminus_real @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_field_def
thf(fact_8622_sinh__field__def,axiom,
    ( sinh_complex
    = ( ^ [Z2: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).

% sinh_field_def
thf(fact_8623_sinh__def,axiom,
    ( sinh_real
    = ( ^ [X4: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ) ).

% sinh_def
thf(fact_8624_sinh__def,axiom,
    ( sinh_complex
    = ( ^ [X4: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) ) ) ) ).

% sinh_def
thf(fact_8625_sinh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( sinh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_8626_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_8627_of__int__round__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_8628_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_8629_of__int__round__ge,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_ge
thf(fact_8630_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_8631_of__int__round__gt,axiom,
    ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_gt
thf(fact_8632_cosh__converges,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) @ zero_zero_real )
      @ ( cosh_real @ X ) ) ).

% cosh_converges
thf(fact_8633_cosh__converges,axiom,
    ! [X: complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_complex @ X @ N3 ) ) @ zero_zero_complex )
      @ ( cosh_complex @ X ) ) ).

% cosh_converges
thf(fact_8634_cot__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_8635_i__even__power,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).

% i_even_power
thf(fact_8636_log__base__10__eq1,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).

% log_base_10_eq1
thf(fact_8637_cot__periodic,axiom,
    ! [X: real] :
      ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cot_real @ X ) ) ).

% cot_periodic
thf(fact_8638_cosh__0,axiom,
    ( ( cosh_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cosh_0
thf(fact_8639_cosh__0,axiom,
    ( ( cosh_real @ zero_zero_real )
    = one_one_real ) ).

% cosh_0
thf(fact_8640_log__one,axiom,
    ! [A: real] :
      ( ( log @ A @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_8641_norm__ii,axiom,
    ( ( real_V1022390504157884413omplex @ imaginary_unit )
    = one_one_real ) ).

% norm_ii
thf(fact_8642_complex__i__mult__minus,axiom,
    ! [X: complex] :
      ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% complex_i_mult_minus
thf(fact_8643_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
          = ( ord_less_real @ one_one_real @ X ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_8644_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
          = ( ord_less_real @ X @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_8645_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
          = ( ord_less_real @ A @ X ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_8646_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
          = ( ord_less_real @ X @ A ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_8647_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
            = ( ord_less_real @ X @ Y ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_8648_log__eq__one,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ A )
          = one_one_real ) ) ) ).

% log_eq_one
thf(fact_8649_divide__i,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).

% divide_i
thf(fact_8650_divide__numeral__i,axiom,
    ! [Z: complex,N: num] :
      ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% divide_numeral_i
thf(fact_8651_i__squared,axiom,
    ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% i_squared
thf(fact_8652_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_8653_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_8654_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ A @ X ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_8655_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
          = ( ord_less_eq_real @ X @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_8656_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
            = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_8657_cot__npi,axiom,
    ! [N: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_8658_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( power_power_real @ A @ B ) )
          = ( semiri5074537144036343181t_real @ B ) ) ) ) ).

% log_pow_cancel
thf(fact_8659_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_8660_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
thf(fact_8661_cosh__real__ge__1,axiom,
    ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).

% cosh_real_ge_1
thf(fact_8662_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( ( times_times_complex @ imaginary_unit @ W )
        = Z )
      = ( W
        = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).

% i_times_eq_iff
thf(fact_8663_log__ln,axiom,
    ( ln_ln_real
    = ( log @ ( exp_real @ one_one_real ) ) ) ).

% log_ln
thf(fact_8664_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_8665_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_8666_sinh__diff,axiom,
    ! [X: real,Y: real] :
      ( ( sinh_real @ ( minus_minus_real @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).

% sinh_diff
thf(fact_8667_cosh__diff,axiom,
    ! [X: real,Y: real] :
      ( ( cosh_real @ ( minus_minus_real @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) ) ) ) ).

% cosh_diff
thf(fact_8668_Complex__eq__i,axiom,
    ! [X: real,Y: real] :
      ( ( ( complex2 @ X @ Y )
        = imaginary_unit )
      = ( ( X = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
thf(fact_8669_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_8670_sinh__plus__cosh,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
      = ( exp_complex @ X ) ) ).

% sinh_plus_cosh
thf(fact_8671_sinh__plus__cosh,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
      = ( exp_real @ X ) ) ).

% sinh_plus_cosh
thf(fact_8672_cosh__plus__sinh,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
      = ( exp_complex @ X ) ) ).

% cosh_plus_sinh
thf(fact_8673_cosh__plus__sinh,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
      = ( exp_real @ X ) ) ).

% cosh_plus_sinh
thf(fact_8674_tanh__def,axiom,
    ( tanh_complex
    = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( sinh_complex @ X4 ) @ ( cosh_complex @ X4 ) ) ) ) ).

% tanh_def
thf(fact_8675_tanh__def,axiom,
    ( tanh_real
    = ( ^ [X4: real] : ( divide_divide_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) ) ) ) ).

% tanh_def
thf(fact_8676_Complex__mult__i,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
      = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).

% Complex_mult_i
thf(fact_8677_i__mult__Complex,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).

% i_mult_Complex
thf(fact_8678_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ B @ X )
          = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).

% log_base_change
thf(fact_8679_log__of__power__eq,axiom,
    ! [M2: nat,B: real,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M2 )
        = ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( semiri5074537144036343181t_real @ N )
          = ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ) ).

% log_of_power_eq
thf(fact_8680_less__log__of__power,axiom,
    ! [B: real,N: nat,M2: real] :
      ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M2 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M2 ) ) ) ) ).

% less_log_of_power
thf(fact_8681_cosh__minus__sinh,axiom,
    ! [X: real] :
      ( ( minus_minus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
      = ( exp_real @ ( uminus_uminus_real @ X ) ) ) ).

% cosh_minus_sinh
thf(fact_8682_cosh__minus__sinh,axiom,
    ! [X: complex] :
      ( ( minus_minus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
      = ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ).

% cosh_minus_sinh
thf(fact_8683_sinh__minus__cosh,axiom,
    ! [X: real] :
      ( ( minus_minus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
      = ( uminus_uminus_real @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ).

% sinh_minus_cosh
thf(fact_8684_sinh__minus__cosh,axiom,
    ! [X: complex] :
      ( ( minus_minus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
      = ( uminus1482373934393186551omplex @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ).

% sinh_minus_cosh
thf(fact_8685_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( times_times_real @ X @ Y ) )
              = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_mult
thf(fact_8686_log__divide,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
              = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_divide
thf(fact_8687_le__log__of__power,axiom,
    ! [B: real,N: nat,M2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M2 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M2 ) ) ) ) ).

% le_log_of_power
thf(fact_8688_log__nat__power,axiom,
    ! [X: real,B: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ B @ ( power_power_real @ X @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).

% log_nat_power
thf(fact_8689_log__inverse,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( log @ A @ ( inverse_inverse_real @ X ) )
            = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).

% log_inverse
thf(fact_8690_cot__def,axiom,
    ( cot_complex
    = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( cos_complex @ X4 ) @ ( sin_complex @ X4 ) ) ) ) ).

% cot_def
thf(fact_8691_cot__def,axiom,
    ( cot_real
    = ( ^ [X4: real] : ( divide_divide_real @ ( cos_real @ X4 ) @ ( sin_real @ X4 ) ) ) ) ).

% cot_def
thf(fact_8692_sinh__double,axiom,
    ! [X: complex] :
      ( ( sinh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sinh_complex @ X ) ) @ ( cosh_complex @ X ) ) ) ).

% sinh_double
thf(fact_8693_sinh__double,axiom,
    ! [X: real] :
      ( ( sinh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sinh_real @ X ) ) @ ( cosh_real @ X ) ) ) ).

% sinh_double
thf(fact_8694_log2__of__power__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( semiri5074537144036343181t_real @ N )
        = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% log2_of_power_eq
thf(fact_8695_log__of__power__less,axiom,
    ! [M2: nat,B: real,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_less
thf(fact_8696_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( B != one_one_real )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( ( log @ A @ X )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_8697_log__of__power__le,axiom,
    ! [M2: nat,B: real,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_le
thf(fact_8698_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_8699_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_8700_less__log2__of__power,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% less_log2_of_power
thf(fact_8701_le__log2__of__power,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% le_log2_of_power
thf(fact_8702_cosh__field__def,axiom,
    ( cosh_real
    = ( ^ [Z2: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z2 ) @ ( exp_real @ ( uminus_uminus_real @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_field_def
thf(fact_8703_cosh__field__def,axiom,
    ( cosh_complex
    = ( ^ [Z2: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).

% cosh_field_def
thf(fact_8704_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_8705_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_8706_hyperbolic__pythagoras,axiom,
    ! [X: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% hyperbolic_pythagoras
thf(fact_8707_hyperbolic__pythagoras,axiom,
    ! [X: real] :
      ( ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% hyperbolic_pythagoras
thf(fact_8708_sinh__square__eq,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).

% sinh_square_eq
thf(fact_8709_sinh__square__eq,axiom,
    ! [X: real] :
      ( ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).

% sinh_square_eq
thf(fact_8710_log2__of__power__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_less
thf(fact_8711_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_8712_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_8713_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_8714_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_8715_cosh__def,axiom,
    ( cosh_real
    = ( ^ [X4: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ) ).

% cosh_def
thf(fact_8716_cosh__def,axiom,
    ( cosh_complex
    = ( ^ [X4: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) ) ) ) ).

% cosh_def
thf(fact_8717_cosh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( cosh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_8718_cot__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).

% cot_gt_zero
thf(fact_8719_log2__of__power__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_le
thf(fact_8720_log__base__10__eq2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).

% log_base_10_eq2
thf(fact_8721_tan__cot_H,axiom,
    ! [X: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
      = ( cot_real @ X ) ) ).

% tan_cot'
thf(fact_8722_Arg__minus__ii,axiom,
    ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
    = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_minus_ii
thf(fact_8723_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K2 )
            & ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_8724_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_8725_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K2 )
     => ( ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_8726_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_8727_of__int__ceiling__cancel,axiom,
    ! [X: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
        = X )
      = ( ? [N3: int] :
            ( X
            = ( ring_1_of_int_rat @ N3 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_8728_of__int__ceiling__cancel,axiom,
    ! [X: real] :
      ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
        = X )
      = ( ? [N3: int] :
            ( X
            = ( ring_1_of_int_real @ N3 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_8729_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_8730_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_8731_ceiling__one,axiom,
    ( ( archim2889992004027027881ng_rat @ one_one_rat )
    = one_one_int ) ).

% ceiling_one
thf(fact_8732_ceiling__one,axiom,
    ( ( archim7802044766580827645g_real @ one_one_real )
    = one_one_int ) ).

% ceiling_one
thf(fact_8733_ceiling__add__of__int,axiom,
    ! [X: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_8734_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_8735_ceiling__diff__of__int,axiom,
    ! [X: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).

% ceiling_diff_of_int
thf(fact_8736_ceiling__diff__of__int,axiom,
    ! [X: real,Z: int] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).

% ceiling_diff_of_int
thf(fact_8737_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_8738_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).

% ceiling_le_zero
thf(fact_8739_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ zero_zero_rat @ X ) ) ).

% zero_less_ceiling
thf(fact_8740_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_8741_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_8742_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_8743_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_8744_ceiling__less__one,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
      = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).

% ceiling_less_one
thf(fact_8745_one__le__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ zero_zero_rat @ X ) ) ).

% one_le_ceiling
thf(fact_8746_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_8747_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_8748_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).

% numeral_less_ceiling
thf(fact_8749_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_8750_ceiling__le__one,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
      = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).

% ceiling_le_one
thf(fact_8751_one__less__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ one_one_rat @ X ) ) ).

% one_less_ceiling
thf(fact_8752_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_8753_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_8754_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_8755_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_8756_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_8757_ceiling__add__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_8758_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_8759_ceiling__diff__numeral,axiom,
    ! [X: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_8760_ceiling__diff__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_8761_ceiling__diff__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_8762_ceiling__diff__one,axiom,
    ! [X: real] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ one_one_real ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_8763_ceiling__numeral__power,axiom,
    ! [X: num,N: nat] :
      ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).

% ceiling_numeral_power
thf(fact_8764_ceiling__numeral__power,axiom,
    ! [X: num,N: nat] :
      ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).

% ceiling_numeral_power
thf(fact_8765_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_8766_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% ceiling_less_zero
thf(fact_8767_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_8768_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).

% zero_le_ceiling
thf(fact_8769_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_divide_eq_div_numeral
thf(fact_8770_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_8771_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% ceiling_less_numeral
thf(fact_8772_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_8773_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_le_ceiling
thf(fact_8774_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_8775_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_8776_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_8777_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).

% neg_numeral_less_ceiling
thf(fact_8778_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_minus_divide_eq_div_numeral
thf(fact_8779_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_8780_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_8781_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_8782_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_le_ceiling
thf(fact_8783_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_8784_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% ceiling_mono
thf(fact_8785_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_8786_le__of__int__ceiling,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% le_of_int_ceiling
thf(fact_8787_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
     => ( ord_less_rat @ X @ Y ) ) ).

% ceiling_less_cancel
thf(fact_8788_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_8789_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_8790_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_8791_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_8792_ceiling__le,axiom,
    ! [X: rat,A: int] :
      ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A ) ) ).

% ceiling_le
thf(fact_8793_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).

% less_ceiling_iff
thf(fact_8794_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_8795_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).

% ceiling_add_le
thf(fact_8796_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_8797_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_8798_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).

% of_int_ceiling_le_add_one
thf(fact_8799_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_8800_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_8801_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% ceiling_divide_eq_div
thf(fact_8802_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] :
      ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% ceiling_divide_eq_div
thf(fact_8803_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_8804_ceiling__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
      & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).

% ceiling_correct
thf(fact_8805_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_8806_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_8807_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_8808_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( ( archim2889992004027027881ng_rat @ X )
        = A )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
        & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_8809_ceiling__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I3 ) ) )
           => ( P @ I3 ) ) ) ) ).

% ceiling_split
thf(fact_8810_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I3 ) ) )
           => ( P @ I3 ) ) ) ) ).

% ceiling_split
thf(fact_8811_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_8812_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_8813_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_8814_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_8815_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% le_ceiling_iff
thf(fact_8816_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_8817_ceiling__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ P2 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_8818_ceiling__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_eq_rat @ P2 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_8819_ceiling__divide__lower,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P2 ) ) ).

% ceiling_divide_lower
thf(fact_8820_ceiling__divide__lower,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P2 ) ) ).

% ceiling_divide_lower
thf(fact_8821_ceiling__eq,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X )
          = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_8822_ceiling__eq,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_8823_cis__minus__pi__half,axiom,
    ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
    = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).

% cis_minus_pi_half
thf(fact_8824_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K2: 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 @ K2 ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K2 ) ) @ X )
            & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_8825_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
            = ( semiri1314217659103216013at_int @ N ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K2 )
            & ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_8826_floor__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K2 )
     => ( ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_8827_arctan__def,axiom,
    ( arctan
    = ( ^ [Y3: real] :
          ( the_real
          @ ^ [X4: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
              & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X4 )
                = Y3 ) ) ) ) ) ).

% arctan_def
thf(fact_8828_powr__one__eq__one,axiom,
    ! [A: real] :
      ( ( powr_real @ one_one_real @ A )
      = one_one_real ) ).

% powr_one_eq_one
thf(fact_8829_of__int__floor__cancel,axiom,
    ! [X: real] :
      ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) )
        = X )
      = ( ? [N3: int] :
            ( X
            = ( ring_1_of_int_real @ N3 ) ) ) ) ).

% of_int_floor_cancel
thf(fact_8830_of__int__floor__cancel,axiom,
    ! [X: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) )
        = X )
      = ( ? [N3: int] :
            ( X
            = ( ring_1_of_int_rat @ N3 ) ) ) ) ).

% of_int_floor_cancel
thf(fact_8831_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_8832_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_8833_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_8834_floor__one,axiom,
    ( ( archim6058952711729229775r_real @ one_one_real )
    = one_one_int ) ).

% floor_one
thf(fact_8835_floor__one,axiom,
    ( ( archim3151403230148437115or_rat @ one_one_rat )
    = one_one_int ) ).

% floor_one
thf(fact_8836_powr__less__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel_iff
thf(fact_8837_norm__cis,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
      = one_one_real ) ).

% norm_cis
thf(fact_8838_cis__zero,axiom,
    ( ( cis @ zero_zero_real )
    = one_one_complex ) ).

% cis_zero
thf(fact_8839_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( powr_real @ A @ X )
          = one_one_real )
        = ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_8840_powr__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ one_one_real )
        = X ) ) ).

% powr_one
thf(fact_8841_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ( powr_real @ X @ one_one_real )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% powr_one_gt_zero_iff
thf(fact_8842_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% powr_le_cancel_iff
thf(fact_8843_numeral__powr__numeral__real,axiom,
    ! [M2: num,N: num] :
      ( ( powr_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( power_power_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_powr_numeral_real
thf(fact_8844_floor__diff__of__int,axiom,
    ! [X: real,Z: int] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ Z ) ) ).

% floor_diff_of_int
thf(fact_8845_floor__diff__of__int,axiom,
    ! [X: rat,Z: int] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ Z ) ) ).

% floor_diff_of_int
thf(fact_8846_cis__pi,axiom,
    ( ( cis @ pi )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% cis_pi
thf(fact_8847_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_8848_zero__le__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).

% zero_le_floor
thf(fact_8849_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_8850_floor__less__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
      = ( ord_less_rat @ X @ zero_zero_rat ) ) ).

% floor_less_zero
thf(fact_8851_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_8852_numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).

% numeral_le_floor
thf(fact_8853_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_8854_zero__less__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% zero_less_floor
thf(fact_8855_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_8856_floor__le__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_le_zero
thf(fact_8857_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_8858_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).

% floor_less_numeral
thf(fact_8859_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_8860_one__le__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% one_le_floor
thf(fact_8861_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_8862_floor__less__one,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_less_one
thf(fact_8863_floor__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% floor_neg_numeral
thf(fact_8864_floor__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% floor_neg_numeral
thf(fact_8865_floor__diff__numeral,axiom,
    ! [X: real,V: num] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_8866_floor__diff__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_8867_floor__diff__one,axiom,
    ! [X: real] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ one_one_real ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_8868_floor__diff__one,axiom,
    ! [X: rat] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_8869_floor__numeral__power,axiom,
    ! [X: num,N: nat] :
      ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).

% floor_numeral_power
thf(fact_8870_floor__numeral__power,axiom,
    ! [X: num,N: nat] :
      ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).

% floor_numeral_power
thf(fact_8871_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( powr_real @ A @ ( log @ A @ X ) )
            = X ) ) ) ) ).

% powr_log_cancel
thf(fact_8872_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( powr_real @ A @ Y ) )
          = Y ) ) ) ).

% log_powr_cancel
thf(fact_8873_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_divide_eq_div_numeral
thf(fact_8874_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
        = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).

% powr_numeral
thf(fact_8875_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_8876_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_less_floor
thf(fact_8877_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_8878_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% floor_le_numeral
thf(fact_8879_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_8880_one__less__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).

% one_less_floor
thf(fact_8881_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_8882_floor__le__one,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_8883_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_8884_neg__numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).

% neg_numeral_le_floor
thf(fact_8885_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_8886_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_8887_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_8888_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).

% floor_one_divide_eq_div_numeral
thf(fact_8889_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_8890_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_divide_eq_div_numeral
thf(fact_8891_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_8892_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_less_floor
thf(fact_8893_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_8894_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% floor_le_neg_numeral
thf(fact_8895_square__powr__half,axiom,
    ! [X: real] :
      ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X ) ) ).

% square_powr_half
thf(fact_8896_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_one_divide_eq_div_numeral
thf(fact_8897_powr__powr,axiom,
    ! [X: real,A: real,B: real] :
      ( ( powr_real @ ( powr_real @ X @ A ) @ B )
      = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).

% powr_powr
thf(fact_8898_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_8899_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).

% floor_mono
thf(fact_8900_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_8901_of__int__floor__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).

% of_int_floor_le
thf(fact_8902_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_8903_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
     => ( ord_less_rat @ X @ Y ) ) ).

% floor_less_cancel
thf(fact_8904_powr__less__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ one_one_real @ X )
       => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).

% powr_less_mono
thf(fact_8905_powr__less__cancel,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
     => ( ( ord_less_real @ one_one_real @ X )
       => ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel
thf(fact_8906_powr__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ one_one_real @ X )
       => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).

% powr_mono
thf(fact_8907_cis__mult,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).

% cis_mult
thf(fact_8908_cis__divide,axiom,
    ! [A: real,B: real] :
      ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).

% cis_divide
thf(fact_8909_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_8910_le__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).

% le_floor_iff
thf(fact_8911_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_8912_floor__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).

% floor_less_iff
thf(fact_8913_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_8914_le__floor__add,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) ) ) ).

% le_floor_add
thf(fact_8915_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).

% gr_one_powr
thf(fact_8916_powr__inj,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ( powr_real @ A @ X )
            = ( powr_real @ A @ Y ) )
          = ( X = Y ) ) ) ) ).

% powr_inj
thf(fact_8917_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_8918_floor__add__int,axiom,
    ! [X: rat,Z: int] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ) ).

% floor_add_int
thf(fact_8919_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_8920_int__add__floor,axiom,
    ! [Z: int,X: rat] :
      ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ) ).

% int_add_floor
thf(fact_8921_powr__le1,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 @ X @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_8922_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ( ord_less_eq_real @ X @ Y )
           => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_8923_ge__one__powr__ge__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_8924_powr__mult,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
          = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mult
thf(fact_8925_floor__divide__of__int__eq,axiom,
    ! [K2: int,L: int] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K2 ) @ ( ring_1_of_int_real @ L ) ) )
      = ( divide_divide_int @ K2 @ L ) ) ).

% floor_divide_of_int_eq
thf(fact_8926_floor__divide__of__int__eq,axiom,
    ! [K2: int,L: int] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K2 ) @ ( ring_1_of_int_rat @ L ) ) )
      = ( divide_divide_int @ K2 @ L ) ) ).

% floor_divide_of_int_eq
thf(fact_8927_floor__power,axiom,
    ! [X: real,N: nat] :
      ( ( X
        = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
     => ( ( archim6058952711729229775r_real @ ( power_power_real @ X @ N ) )
        = ( power_power_int @ ( archim6058952711729229775r_real @ X ) @ N ) ) ) ).

% floor_power
thf(fact_8928_floor__power,axiom,
    ! [X: rat,N: nat] :
      ( ( X
        = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
     => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X @ N ) )
        = ( power_power_int @ ( archim3151403230148437115or_rat @ X ) @ N ) ) ) ).

% floor_power
thf(fact_8929_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
      = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).

% divide_powr_uminus
thf(fact_8930_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).

% ln_powr
thf(fact_8931_log__powr,axiom,
    ! [X: real,B: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( log @ B @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( log @ B @ X ) ) ) ) ).

% log_powr
thf(fact_8932_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_8933_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] :
      ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
      = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).

% powr_diff
thf(fact_8934_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_8935_one__add__floor,axiom,
    ! [X: rat] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).

% one_add_floor
thf(fact_8936_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K2: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
            = K2 )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K2 ) ) @ X )
            & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K2 @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_8937_floor__divide__of__nat__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_8938_floor__divide__of__nat__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_8939_powr__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
          = ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).

% powr_less_iff
thf(fact_8940_less__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
          = ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).

% less_powr_iff
thf(fact_8941_log__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ B @ X ) @ Y )
          = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_less_iff
thf(fact_8942_less__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ Y @ ( log @ B @ X ) )
          = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).

% less_log_iff
thf(fact_8943_ceiling__altdef,axiom,
    ( archim7802044766580827645g_real
    = ( ^ [X4: real] :
          ( if_int
          @ ( X4
            = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) )
          @ ( archim6058952711729229775r_real @ X4 )
          @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int ) ) ) ) ).

% ceiling_altdef
thf(fact_8944_ceiling__altdef,axiom,
    ( archim2889992004027027881ng_rat
    = ( ^ [X4: rat] :
          ( if_int
          @ ( X4
            = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) )
          @ ( archim3151403230148437115or_rat @ X4 )
          @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int ) ) ) ) ).

% ceiling_altdef
thf(fact_8945_ceiling__diff__floor__le__1,axiom,
    ! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_8946_ceiling__diff__floor__le__1,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_8947_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_8948_floor__eq,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N ) ) ) ).

% floor_eq
thf(fact_8949_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_8950_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_8951_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_8952_DeMoivre,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_complex @ ( cis @ A ) @ N )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).

% DeMoivre
thf(fact_8953_floor__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim6058952711729229775r_real @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I3 ) @ T )
              & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) ) )
           => ( P @ I3 ) ) ) ) ).

% floor_split
thf(fact_8954_floor__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim3151403230148437115or_rat @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I3 ) @ T )
              & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) ) )
           => ( P @ I3 ) ) ) ) ).

% floor_split
thf(fact_8955_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_8956_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( ( archim3151403230148437115or_rat @ X )
        = A )
      = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).

% floor_eq_iff
thf(fact_8957_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_8958_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
     => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X )
          = Z ) ) ) ).

% floor_unique
thf(fact_8959_powr__minus__divide,axiom,
    ! [X: real,A: real] :
      ( ( powr_real @ X @ ( uminus_uminus_real @ A ) )
      = ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ).

% powr_minus_divide
thf(fact_8960_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_8961_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_8962_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_8963_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% less_floor_iff
thf(fact_8964_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_8965_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_8966_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_8967_floor__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_8968_powr__neg__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X ) ) ) ).

% powr_neg_one
thf(fact_8969_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
        = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).

% powr_mult_base
thf(fact_8970_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
          = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).

% le_log_iff
thf(fact_8971_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
          = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_le_iff
thf(fact_8972_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
          = ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).

% le_powr_iff
thf(fact_8973_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
          = ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).

% powr_le_iff
thf(fact_8974_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N ) ) ) ).

% floor_eq2
thf(fact_8975_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
        = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).

% floor_divide_real_eq_div
thf(fact_8976_floor__divide__lower,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ Q3 ) @ P2 ) ) ).

% floor_divide_lower
thf(fact_8977_floor__divide__lower,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ Q3 ) @ P2 ) ) ).

% floor_divide_lower
thf(fact_8978_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_8979_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).

% ln_powr_bound2
thf(fact_8980_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ ( log @ B @ X ) @ Y )
            = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_8981_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
            = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_8982_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
            = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_8983_powr__def,axiom,
    ( powr_real
    = ( ^ [X4: real,A5: real] : ( if_real @ ( X4 = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A5 @ ( ln_ln_real @ X4 ) ) ) ) ) ) ).

% powr_def
thf(fact_8984_floor__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_real @ P2 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) ) ) ).

% floor_divide_upper
thf(fact_8985_floor__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_rat @ P2 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) ) ) ).

% floor_divide_upper
thf(fact_8986_round__def,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X4: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_8987_round__def,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X4: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_8988_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ ( log @ B @ X ) @ Y )
            = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_8989_powr__half__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X ) ) ) ).

% powr_half_sqrt
thf(fact_8990_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_8991_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X4 )
          & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X4 )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_8992_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
            & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X4 )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_8993_floor__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_8994_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y3: real] :
          ( the_real
          @ ^ [X4: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
              & ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X4 )
                = Y3 ) ) ) ) ) ).

% arcsin_def
thf(fact_8995_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( bij_betw_nat_complex
        @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        @ ( set_ord_lessThan_nat @ N )
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_8996_arcosh__def,axiom,
    ( arcosh_real
    = ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arcosh_def
thf(fact_8997_round__altdef,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X4: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X4 ) ) @ ( archim7802044766580827645g_real @ X4 ) @ ( archim6058952711729229775r_real @ X4 ) ) ) ) ).

% round_altdef
thf(fact_8998_round__altdef,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X4: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X4 ) ) @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim3151403230148437115or_rat @ X4 ) ) ) ) ).

% round_altdef
thf(fact_8999_The__split__eq,axiom,
    ! [X: code_integer,Y: $o] :
      ( ( the_Pr406357557219058217eger_o
        @ ( produc7828578312038201481er_o_o
          @ ^ [X10: code_integer,Y7: $o] :
              ( ( X = X10 )
              & ( Y = Y7 ) ) ) )
      = ( produc6677183202524767010eger_o @ X @ Y ) ) ).

% The_split_eq
thf(fact_9000_The__split__eq,axiom,
    ! [X: num,Y: num] :
      ( ( the_Pr6395592806576110876um_num
        @ ( produc5703948589228662326_num_o
          @ ^ [X10: num,Y7: num] :
              ( ( X = X10 )
              & ( Y = Y7 ) ) ) )
      = ( product_Pair_num_num @ X @ Y ) ) ).

% The_split_eq
thf(fact_9001_The__split__eq,axiom,
    ! [X: nat,Y: num] :
      ( ( the_Pr8265262403268641490at_num
        @ ( produc4927758841916487424_num_o
          @ ^ [X10: nat,Y7: num] :
              ( ( X = X10 )
              & ( Y = Y7 ) ) ) )
      = ( product_Pair_nat_num @ X @ Y ) ) ).

% The_split_eq
thf(fact_9002_The__split__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( the_Pr7557018466319803784at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X10: nat,Y7: nat] :
              ( ( X = X10 )
              & ( Y = Y7 ) ) ) )
      = ( product_Pair_nat_nat @ X @ Y ) ) ).

% The_split_eq
thf(fact_9003_The__split__eq,axiom,
    ! [X: int,Y: int] :
      ( ( the_Pr4378521158711661632nt_int
        @ ( produc4947309494688390418_int_o
          @ ^ [X10: int,Y7: int] :
              ( ( X = X10 )
              & ( Y = Y7 ) ) ) )
      = ( product_Pair_int_int @ X @ Y ) ) ).

% The_split_eq
thf(fact_9004_of__real__1,axiom,
    ( ( real_V1803761363581548252l_real @ one_one_real )
    = one_one_real ) ).

% of_real_1
thf(fact_9005_of__real__1,axiom,
    ( ( real_V4546457046886955230omplex @ one_one_real )
    = one_one_complex ) ).

% of_real_1
thf(fact_9006_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( real_V1803761363581548252l_real @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_9007_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( real_V4546457046886955230omplex @ X )
        = one_one_complex )
      = ( X = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_9008_of__real__mult,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1803761363581548252l_real @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).

% of_real_mult
thf(fact_9009_of__real__mult,axiom,
    ! [X: real,Y: real] :
      ( ( real_V4546457046886955230omplex @ ( times_times_real @ X @ Y ) )
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).

% of_real_mult
thf(fact_9010_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% of_real_numeral
thf(fact_9011_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
      = ( numera6690914467698888265omplex @ W ) ) ).

% of_real_numeral
thf(fact_9012_of__real__divide,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
      = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).

% of_real_divide
thf(fact_9013_of__real__divide,axiom,
    ! [X: real,Y: real] :
      ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
      = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).

% of_real_divide
thf(fact_9014_of__real__power,axiom,
    ! [X: real,N: nat] :
      ( ( real_V1803761363581548252l_real @ ( power_power_real @ X @ N ) )
      = ( power_power_real @ ( real_V1803761363581548252l_real @ X ) @ N ) ) ).

% of_real_power
thf(fact_9015_of__real__power,axiom,
    ! [X: real,N: nat] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ X @ N ) )
      = ( power_power_complex @ ( real_V4546457046886955230omplex @ X ) @ N ) ) ).

% of_real_power
thf(fact_9016_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_9017_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_9018_of__real__diff,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1803761363581548252l_real @ ( minus_minus_real @ X @ Y ) )
      = ( minus_minus_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).

% of_real_diff
thf(fact_9019_of__real__diff,axiom,
    ! [X: real,Y: real] :
      ( ( real_V4546457046886955230omplex @ ( minus_minus_real @ X @ Y ) )
      = ( minus_minus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).

% of_real_diff
thf(fact_9020_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_9021_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_9022_cos__of__real__pi,axiom,
    ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_of_real_pi
thf(fact_9023_cos__of__real__pi,axiom,
    ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% cos_of_real_pi
thf(fact_9024_exp__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i'
thf(fact_9025_exp__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i
thf(fact_9026_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_9027_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_9028_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_9029_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_9030_exp__two__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
    = one_one_complex ) ).

% exp_two_pi_i'
thf(fact_9031_exp__two__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
    = one_one_complex ) ).

% exp_two_pi_i
thf(fact_9032_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_9033_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_9034_sin__of__real__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_of_real_pi_half
thf(fact_9035_sin__of__real__pi__half,axiom,
    ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = one_one_complex ) ).

% sin_of_real_pi_half
thf(fact_9036_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A3: complex,R4: real] :
      ( Z
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R4 ) @ ( exp_complex @ A3 ) ) ) ).

% complex_exp_exists
thf(fact_9037_scaleR__conv__of__real,axiom,
    ( real_V1485227260804924795R_real
    = ( ^ [R5: real] : ( times_times_real @ ( real_V1803761363581548252l_real @ R5 ) ) ) ) ).

% scaleR_conv_of_real
thf(fact_9038_scaleR__conv__of__real,axiom,
    ( real_V2046097035970521341omplex
    = ( ^ [R5: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) ) ) ) ).

% scaleR_conv_of_real
thf(fact_9039_of__real__def,axiom,
    ( real_V1803761363581548252l_real
    = ( ^ [R5: real] : ( real_V1485227260804924795R_real @ R5 @ one_one_real ) ) ) ).

% of_real_def
thf(fact_9040_of__real__def,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [R5: real] : ( real_V2046097035970521341omplex @ R5 @ one_one_complex ) ) ) ).

% of_real_def
thf(fact_9041_frac__ge__0,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).

% frac_ge_0
thf(fact_9042_frac__ge__0,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).

% frac_ge_0
thf(fact_9043_frac__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).

% frac_lt_1
thf(fact_9044_frac__lt__1,axiom,
    ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_9045_frac__1__eq,axiom,
    ! [X: real] :
      ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
      = ( archim2898591450579166408c_real @ X ) ) ).

% frac_1_eq
thf(fact_9046_frac__1__eq,axiom,
    ! [X: rat] :
      ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
      = ( archimedean_frac_rat @ X ) ) ).

% frac_1_eq
thf(fact_9047_nonzero__of__real__divide,axiom,
    ! [Y: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
        = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ) ).

% nonzero_of_real_divide
thf(fact_9048_nonzero__of__real__divide,axiom,
    ! [Y: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
        = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ) ).

% nonzero_of_real_divide
thf(fact_9049_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R2: real] :
      ( ( times_times_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ ( times_times_real @ X @ R2 ) @ ( times_times_real @ Y @ R2 ) ) ) ).

% Complex_mult_complex_of_real
thf(fact_9050_complex__of__real__mult__Complex,axiom,
    ! [R2: real,X: real,Y: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X @ Y ) )
      = ( complex2 @ ( times_times_real @ R2 @ X ) @ ( times_times_real @ R2 @ Y ) ) ) ).

% complex_of_real_mult_Complex
thf(fact_9051_cis__conv__exp,axiom,
    ( cis
    = ( ^ [B5: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B5 ) ) ) ) ) ).

% cis_conv_exp
thf(fact_9052_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_9053_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_9054_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_int,T4: set_nat,H2: int > nat,S3: set_int,T3: set_nat,G: nat > nat] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_nat @ T4 )
       => ( ( bij_betw_int_nat @ H2 @ ( minus_minus_set_int @ S3 @ S4 ) @ ( minus_minus_set_nat @ T3 @ T4 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ S4 )
               => ( ( G @ ( H2 @ A3 ) )
                  = zero_zero_nat ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ T4 )
                 => ( ( G @ B3 )
                    = zero_zero_nat ) )
             => ( ( groups4541462559716669496nt_nat
                  @ ^ [X4: int] : ( G @ ( H2 @ X4 ) )
                  @ S3 )
                = ( groups3542108847815614940at_nat @ G @ T3 ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_9055_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_real,T4: set_nat,H2: real > nat,S3: set_real,T3: set_nat,G: nat > nat] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_nat @ T4 )
       => ( ( bij_betw_real_nat @ H2 @ ( minus_minus_set_real @ S3 @ S4 ) @ ( minus_minus_set_nat @ T3 @ T4 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ S4 )
               => ( ( G @ ( H2 @ A3 ) )
                  = zero_zero_nat ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ T4 )
                 => ( ( G @ B3 )
                    = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat
                  @ ^ [X4: real] : ( G @ ( H2 @ X4 ) )
                  @ S3 )
                = ( groups3542108847815614940at_nat @ G @ T3 ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_9056_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_complex,T4: set_nat,H2: complex > nat,S3: set_complex,T3: set_nat,G: nat > nat] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( finite_finite_nat @ T4 )
       => ( ( bij_betw_complex_nat @ H2 @ ( minus_811609699411566653omplex @ S3 @ S4 ) @ ( minus_minus_set_nat @ T3 @ T4 ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ S4 )
               => ( ( G @ ( H2 @ A3 ) )
                  = zero_zero_nat ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ T4 )
                 => ( ( G @ B3 )
                    = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat
                  @ ^ [X4: complex] : ( G @ ( H2 @ X4 ) )
                  @ S3 )
                = ( groups3542108847815614940at_nat @ G @ T3 ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_9057_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_int,T4: set_nat,H2: int > nat,S3: set_int,T3: set_nat,G: nat > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_nat @ T4 )
       => ( ( bij_betw_int_nat @ H2 @ ( minus_minus_set_int @ S3 @ S4 ) @ ( minus_minus_set_nat @ T3 @ T4 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ S4 )
               => ( ( G @ ( H2 @ A3 ) )
                  = zero_zero_real ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ T4 )
                 => ( ( G @ B3 )
                    = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real
                  @ ^ [X4: int] : ( G @ ( H2 @ X4 ) )
                  @ S3 )
                = ( groups6591440286371151544t_real @ G @ T3 ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_9058_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_real,T4: set_nat,H2: real > nat,S3: set_real,T3: set_nat,G: nat > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_nat @ T4 )
       => ( ( bij_betw_real_nat @ H2 @ ( minus_minus_set_real @ S3 @ S4 ) @ ( minus_minus_set_nat @ T3 @ T4 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ S4 )
               => ( ( G @ ( H2 @ A3 ) )
                  = zero_zero_real ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ T4 )
                 => ( ( G @ B3 )
                    = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real
                  @ ^ [X4: real] : ( G @ ( H2 @ X4 ) )
                  @ S3 )
                = ( groups6591440286371151544t_real @ G @ T3 ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_9059_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_complex,T4: set_nat,H2: complex > nat,S3: set_complex,T3: set_nat,G: nat > real] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( finite_finite_nat @ T4 )
       => ( ( bij_betw_complex_nat @ H2 @ ( minus_811609699411566653omplex @ S3 @ S4 ) @ ( minus_minus_set_nat @ T3 @ T4 ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ S4 )
               => ( ( G @ ( H2 @ A3 ) )
                  = zero_zero_real ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ T4 )
                 => ( ( G @ B3 )
                    = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real
                  @ ^ [X4: complex] : ( G @ ( H2 @ X4 ) )
                  @ S3 )
                = ( groups6591440286371151544t_real @ G @ T3 ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_9060_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_int,T4: set_real,H2: int > real,S3: set_int,T3: set_real,G: real > int] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ( bij_betw_int_real @ H2 @ ( minus_minus_set_int @ S3 @ S4 ) @ ( minus_minus_set_real @ T3 @ T4 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ S4 )
               => ( ( G @ ( H2 @ A3 ) )
                  = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ T4 )
                 => ( ( G @ B3 )
                    = zero_zero_int ) )
             => ( ( groups4538972089207619220nt_int
                  @ ^ [X4: int] : ( G @ ( H2 @ X4 ) )
                  @ S3 )
                = ( groups1932886352136224148al_int @ G @ T3 ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_9061_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_int,T4: set_complex,H2: int > complex,S3: set_int,T3: set_complex,G: complex > int] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ( bij_betw_int_complex @ H2 @ ( minus_minus_set_int @ S3 @ S4 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ S4 )
               => ( ( G @ ( H2 @ A3 ) )
                  = zero_zero_int ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ T4 )
                 => ( ( G @ B3 )
                    = zero_zero_int ) )
             => ( ( groups4538972089207619220nt_int
                  @ ^ [X4: int] : ( G @ ( H2 @ X4 ) )
                  @ S3 )
                = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_9062_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_int,T4: set_nat,H2: int > nat,S3: set_int,T3: set_nat,G: nat > int] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_nat @ T4 )
       => ( ( bij_betw_int_nat @ H2 @ ( minus_minus_set_int @ S3 @ S4 ) @ ( minus_minus_set_nat @ T3 @ T4 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ S4 )
               => ( ( G @ ( H2 @ A3 ) )
                  = zero_zero_int ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ T4 )
                 => ( ( G @ B3 )
                    = zero_zero_int ) )
             => ( ( groups4538972089207619220nt_int
                  @ ^ [X4: int] : ( G @ ( H2 @ X4 ) )
                  @ S3 )
                = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_9063_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [S4: set_int,T4: set_int,H2: int > int,S3: set_int,T3: set_int,G: int > int] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ( bij_betw_int_int @ H2 @ ( minus_minus_set_int @ S3 @ S4 ) @ ( minus_minus_set_int @ T3 @ T4 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ S4 )
               => ( ( G @ ( H2 @ A3 ) )
                  = zero_zero_int ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ T4 )
                 => ( ( G @ B3 )
                    = zero_zero_int ) )
             => ( ( groups4538972089207619220nt_int
                  @ ^ [X4: int] : ( G @ ( H2 @ X4 ) )
                  @ S3 )
                = ( groups4538972089207619220nt_int @ G @ T3 ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
thf(fact_9064_i__complex__of__real,axiom,
    ! [R2: real] :
      ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ zero_zero_real @ R2 ) ) ).

% i_complex_of_real
thf(fact_9065_complex__of__real__i,axiom,
    ! [R2: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
      = ( complex2 @ zero_zero_real @ R2 ) ) ).

% complex_of_real_i
thf(fact_9066_Complex__eq,axiom,
    ( complex2
    = ( ^ [A5: real,B5: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A5 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B5 ) ) ) ) ) ).

% Complex_eq
thf(fact_9067_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R4: real,A3: real] :
      ( Z
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R4 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A3 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A3 ) ) ) ) ) ) ).

% complex_split_polar
thf(fact_9068_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X4: real] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X4 )
              & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9069_cmod__unit__one,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
      = one_one_real ) ).

% cmod_unit_one
thf(fact_9070_cmod__complex__polar,axiom,
    ! [R2: real,A: real] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
      = ( abs_abs_real @ R2 ) ) ).

% cmod_complex_polar
thf(fact_9071_csqrt__ii,axiom,
    ( ( csqrt @ imaginary_unit )
    = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt_ii
thf(fact_9072_modulo__int__unfold,axiom,
    ! [L: int,K2: int,N: nat,M2: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K2 )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K2 )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K2 )
             != ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N @ M2 ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_9073_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_9074_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% mask_nat_positive_iff
thf(fact_9075_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( ( csqrt @ Z )
        = one_one_complex )
      = ( Z = one_one_complex ) ) ).

% csqrt_eq_1
thf(fact_9076_csqrt__1,axiom,
    ( ( csqrt @ one_one_complex )
    = one_one_complex ) ).

% csqrt_1
thf(fact_9077_dvd__mult__sgn__iff,axiom,
    ! [L: int,K2: int,R2: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ K2 @ ( sgn_sgn_int @ R2 ) ) )
      = ( ( dvd_dvd_int @ L @ K2 )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_9078_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K2: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K2 ) )
      = ( ( dvd_dvd_int @ L @ K2 )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_9079_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K2 )
      = ( ( dvd_dvd_int @ L @ K2 )
        & ( ( R2 = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_9080_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K2 )
      = ( ( dvd_dvd_int @ L @ K2 )
        & ( ( R2 = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_9081_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_9082_less__eq__mask,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).

% less_eq_mask
thf(fact_9083_int__sgnE,axiom,
    ! [K2: int] :
      ~ ! [N2: nat,L4: int] :
          ( K2
         != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% int_sgnE
thf(fact_9084_mask__nonnegative__int,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).

% mask_nonnegative_int
thf(fact_9085_not__mask__negative__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_9086_sgn__mod,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ L @ K2 )
       => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K2 @ L ) )
          = ( sgn_sgn_int @ L ) ) ) ) ).

% sgn_mod
thf(fact_9087_less__mask,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).

% less_mask
thf(fact_9088_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_9089_div__sgn__abs__cancel,axiom,
    ! [V: int,K2: int,L: int] :
      ( ( V != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_9090_div__dvd__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( ( dvd_dvd_int @ L @ K2 )
     => ( ( divide_divide_int @ K2 @ L )
        = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_dvd_sgn_abs
thf(fact_9091_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_9092_Suc__mask__eq__exp,axiom,
    ! [N: nat] :
      ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_mask_eq_exp
thf(fact_9093_mask__nat__less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% mask_nat_less_exp
thf(fact_9094_div__noneq__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => ( ( ( sgn_sgn_int @ K2 )
         != ( sgn_sgn_int @ L ) )
       => ( ( divide_divide_int @ K2 @ L )
          = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L ) ) )
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( dvd_dvd_int @ L @ K2 ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_9095_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K2: int,Q3: int] :
      ( ( ( sgn_sgn_int @ R2 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
       => ( ( K2
            = ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R2 ) )
         => ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q3 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_9096_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_9097_mask__half__int,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% mask_half_int
thf(fact_9098_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_9099_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A12 = K3 )
              & ( A23 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L2: int,K3: int,Q4: int] :
              ( ( A12 = K3 )
              & ( A23 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q4 @ L2 ) ) )
          | ? [R5: int,L2: int,K3: int,Q4: int] :
              ( ( A12 = K3 )
              & ( A23 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L2 ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q4 @ L2 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_9100_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A1 @ A22 @ A33 )
     => ( ( ( A22 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
       => ( ! [Q2: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
             => ( ( A22 != zero_zero_int )
               => ( A1
                 != ( times_times_int @ Q2 @ A22 ) ) ) )
         => ~ ! [R4: int,Q2: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q2 @ R4 ) )
               => ( ( ( sgn_sgn_int @ R4 )
                    = ( sgn_sgn_int @ A22 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ A22 ) )
                   => ( A1
                     != ( plus_plus_int @ ( times_times_int @ Q2 @ A22 ) @ R4 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_9101_num_Osize__gen_I2_J,axiom,
    ! [X2: num] :
      ( ( size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_9102_divide__int__unfold,axiom,
    ! [L: int,K2: int,N: nat,M2: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K2 )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K2 )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) ) ) )
          & ( ( ( sgn_sgn_int @ K2 )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M2 @ N )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_9103_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X4: rat] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X4 )
              & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9104_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9105_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K3: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9106_powr__int,axiom,
    ! [X: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_9107_nat__numeral,axiom,
    ! [K2: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_nat @ K2 ) ) ).

% nat_numeral
thf(fact_9108_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_9109_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = ( root @ N @ Y ) )
        = ( X = Y ) ) ) ).

% real_root_eq_iff
thf(fact_9110_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_9111_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_9112_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% real_root_less_iff
thf(fact_9113_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% real_root_le_iff
thf(fact_9114_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
      = ( ( ord_less_int @ zero_zero_int @ Z )
        & ( ord_less_int @ W @ Z ) ) ) ).

% zless_nat_conj
thf(fact_9115_real__root__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_9116_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = one_one_real )
        = ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_9117_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_9118_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9119_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9120_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_9121_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_9122_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_9123_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9124_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9125_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_9126_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_9127_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( nat2 @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_9128_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = ( nat2 @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_9129_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_9130_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_9131_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos2
thf(fact_9132_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_9133_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_9134_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_9135_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_9136_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_9137_nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( bit_se2002935070580805687sk_nat @ N ) ) ).

% nat_mask_eq
thf(fact_9138_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( sgn_sgn_real @ ( root @ N @ X ) )
        = ( sgn_sgn_real @ X ) ) ) ).

% sgn_root
thf(fact_9139_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A5: rat] : ( if_rat @ ( A5 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A5 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9140_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R2 )
     => ~ ! [S: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S )
           => ! [T5: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T5 )
               => ( R2
                 != ( plus_plus_rat @ S @ T5 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9141_real__root__mult,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( root @ N @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).

% real_root_mult
thf(fact_9142_real__root__mult__exp,axiom,
    ! [M2: nat,N: nat,X: real] :
      ( ( root @ ( times_times_nat @ M2 @ N ) @ X )
      = ( root @ M2 @ ( root @ N @ X ) ) ) ).

% real_root_mult_exp
thf(fact_9143_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I3: num] : ( nat2 @ ( numeral_numeral_int @ I3 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_9144_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_9145_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_9146_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
        = X ) ) ).

% sgn_power_root
thf(fact_9147_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_9148_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M3 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_9149_split__root,axiom,
    ! [P: real > $o,N: nat,X: real] :
      ( ( P @ ( root @ N @ X ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N )
         => ! [Y3: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) )
                = X )
             => ( P @ Y3 ) ) ) ) ) ).

% split_root
thf(fact_9150_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_9151_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_9152_real__root__power,axiom,
    ! [N: nat,X: real,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( power_power_real @ X @ K2 ) )
        = ( power_power_real @ ( root @ N @ X ) @ K2 ) ) ) ).

% real_root_power
thf(fact_9153_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( abs_abs_real @ X ) )
        = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).

% real_root_abs
thf(fact_9154_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_mono_iff
thf(fact_9155_zless__nat__eq__int__zless,axiom,
    ! [M2: nat,Z: int] :
      ( ( ord_less_nat @ M2 @ ( nat2 @ Z ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z ) ) ).

% zless_nat_eq_int_zless
thf(fact_9156_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
      = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_le_iff
thf(fact_9157_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_9158_int__minus,axiom,
    ! [N: nat,M2: nat] :
      ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M2 ) )
      = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ) ).

% int_minus
thf(fact_9159_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_9160_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_9161_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_9162_real__nat__ceiling__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_9163_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_9164_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_9165_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_gt_zero
thf(fact_9166_real__root__strict__decreasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ( ord_less_real @ one_one_real @ X )
         => ( ord_less_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9167_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9168_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A5: real] : ( if_real @ ( A5 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A5 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_9169_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_9170_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_less_eq_zless
thf(fact_9171_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W )
        | ( ord_less_eq_int @ zero_zero_int @ Z ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_eq_int @ W @ Z ) ) ) ).

% nat_le_eq_zle
thf(fact_9172_le__nat__iff,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( ord_less_eq_nat @ N @ ( nat2 @ K2 ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K2 ) ) ) ).

% le_nat_iff
thf(fact_9173_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_9174_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).

% nat_mult_distrib
thf(fact_9175_Suc__as__int,axiom,
    ( suc
    = ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_9176_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
     => ( ( ord_less_eq_int @ Z6 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_9177_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_9178_nat__abs__triangle__ineq,axiom,
    ! [K2: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K2 @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_9179_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib
thf(fact_9180_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib'
thf(fact_9181_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).

% nat_power_eq
thf(fact_9182_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_9183_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
          = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_9184_div__abs__eq__div__nat,axiom,
    ! [K2: int,L: int] :
      ( ( divide_divide_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_9185_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq3
thf(fact_9186_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_9187_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_pos_pos
thf(fact_9188_real__root__strict__increasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ X @ one_one_real )
           => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9189_real__root__decreasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ord_less_eq_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9190_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos
thf(fact_9191_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( root @ N @ ( power_power_real @ X @ N ) )
          = X ) ) ) ).

% real_root_power_cancel
thf(fact_9192_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N )
            = X )
         => ( ( root @ N @ X )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_9193_odd__real__root__pow,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( root @ N @ X ) @ N )
        = X ) ) ).

% odd_real_root_pow
thf(fact_9194_odd__real__root__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ( power_power_real @ Y @ N )
          = X )
       => ( ( root @ N @ X )
          = Y ) ) ) ).

% odd_real_root_unique
thf(fact_9195_odd__real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( root @ N @ ( power_power_real @ X @ N ) )
        = X ) ) ).

% odd_real_root_power_cancel
thf(fact_9196_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_9197_sgn__power__injE,axiom,
    ! [A: real,N: nat,X: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
        = X )
     => ( ( X
          = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( A = B ) ) ) ) ).

% sgn_power_injE
thf(fact_9198_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_9199_nat__less__iff,axiom,
    ! [W: int,M2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ M2 )
        = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).

% nat_less_iff
thf(fact_9200_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_9201_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_9202_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq4
thf(fact_9203_diff__nat__eq__if,axiom,
    ! [Z6: int,Z: int] :
      ( ( ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_9204_real__root__increasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ X @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9205_ln__root,axiom,
    ! [N: nat,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ln_ln_real @ ( root @ N @ B ) )
          = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% ln_root
thf(fact_9206_log__root,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B @ ( root @ N @ A ) )
          = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_root
thf(fact_9207_log__base__root,axiom,
    ! [N: nat,B: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log @ ( root @ N @ B ) @ X )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).

% log_base_root
thf(fact_9208_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( root @ N @ X )
          = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9209_even__nat__iff,axiom,
    ! [K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K2 ) )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) ) ).

% even_nat_iff
thf(fact_9210_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K3: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ K3
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L2 )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L2 @ K3 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_9211_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
        = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).

% arctan_inverse
thf(fact_9212_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( K3 = zero_zero_int )
            | ( L2 = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            @ L2
            @ ( if_int
              @ ( L2
                = ( uminus_uminus_int @ one_one_int ) )
              @ K3
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_9213_concat__bit__of__zero__2,axiom,
    ! [N: nat,K2: int] :
      ( ( bit_concat_bit @ N @ K2 @ zero_zero_int )
      = ( bit_se2923211474154528505it_int @ N @ K2 ) ) ).

% concat_bit_of_zero_2
thf(fact_9214_and__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K2 @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K2 )
        | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% and_nonnegative_int_iff
thf(fact_9215_and__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K2 @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K2 @ zero_zero_int )
        & ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_9216_take__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_Suc_0
thf(fact_9217_and__minus__numerals_I6_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
      = one_one_int ) ).

% and_minus_numerals(6)
thf(fact_9218_and__minus__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = one_one_int ) ).

% and_minus_numerals(2)
thf(fact_9219_and__minus__numerals_I1_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = zero_zero_int ) ).

% and_minus_numerals(1)
thf(fact_9220_and__minus__numerals_I5_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_minus_numerals(5)
thf(fact_9221_diff__rat__def,axiom,
    ( minus_minus_rat
    = ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).

% diff_rat_def
thf(fact_9222_take__bit__tightened__less__eq__nat,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M2 @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N @ Q3 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_9223_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ M2 ) ).

% take_bit_nat_less_eq_self
thf(fact_9224_nat__take__bit__eq,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K2 ) )
        = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K2 ) ) ) ) ).

% nat_take_bit_eq
thf(fact_9225_take__bit__nat__eq,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K2 ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ) ).

% take_bit_nat_eq
thf(fact_9226_take__bit__diff,axiom,
    ! [N: nat,K2: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K2 @ L ) ) ) ).

% take_bit_diff
thf(fact_9227_take__bit__mult,axiom,
    ! [N: nat,K2: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K2 @ L ) ) ) ).

% take_bit_mult
thf(fact_9228_take__bit__minus,axiom,
    ! [N: nat,K2: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K2 ) ) ) ).

% take_bit_minus
thf(fact_9229_concat__bit__take__bit__eq,axiom,
    ! [N: nat,B: int] :
      ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B ) )
      = ( bit_concat_bit @ N @ B ) ) ).

% concat_bit_take_bit_eq
thf(fact_9230_concat__bit__eq__iff,axiom,
    ! [N: nat,K2: int,L: int,R2: int,S2: int] :
      ( ( ( bit_concat_bit @ N @ K2 @ L )
        = ( bit_concat_bit @ N @ R2 @ S2 ) )
      = ( ( ( bit_se2923211474154528505it_int @ N @ K2 )
          = ( bit_se2923211474154528505it_int @ N @ R2 ) )
        & ( L = S2 ) ) ) ).

% concat_bit_eq_iff
thf(fact_9231_take__bit__tightened__less__eq__int,axiom,
    ! [M2: nat,N: nat,K2: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M2 @ K2 ) @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_9232_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_9233_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_9234_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).

% AND_upper2
thf(fact_9235_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).

% AND_upper1
thf(fact_9236_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).

% AND_lower
thf(fact_9237_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ K2 )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_9238_take__bit__nonnegative,axiom,
    ! [N: nat,K2: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ).

% take_bit_nonnegative
thf(fact_9239_take__bit__int__greater__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_int @ K2 @ ( bit_se2923211474154528505it_int @ N @ K2 ) )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_9240_not__take__bit__negative,axiom,
    ! [N: nat,K2: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_9241_pow_Osimps_I1_J,axiom,
    ! [X: num] :
      ( ( pow @ X @ one )
      = X ) ).

% pow.simps(1)
thf(fact_9242_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_9243_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_9244_and__less__eq,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K2 @ L ) @ K2 ) ) ).

% and_less_eq
thf(fact_9245_take__bit__decr__eq,axiom,
    ! [N: nat,K2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K2 )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K2 @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_9246_even__and__iff__int,axiom,
    ! [K2: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K2 @ L ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).

% even_and_iff_int
thf(fact_9247_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K2 )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K2 @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_9248_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ M2 )
        = M2 )
      = ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_9249_take__bit__nat__less__exp,axiom,
    ! [N: nat,M2: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_nat_less_exp
thf(fact_9250_take__bit__nat__eq__self,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( bit_se2925701944663578781it_nat @ N @ M2 )
        = M2 ) ) ).

% take_bit_nat_eq_self
thf(fact_9251_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,M3: nat] : ( modulo_modulo_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_nat_def
thf(fact_9252_take__bit__int__less__exp,axiom,
    ! [N: nat,K2: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_int_less_exp
thf(fact_9253_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_int_def
thf(fact_9254_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ M2 )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 ) ) ).

% take_bit_nat_less_self_iff
thf(fact_9255_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_9256_take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_eq_int @ K2 @ ( bit_se2923211474154528505it_int @ N @ K2 ) )
      = ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_9257_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ K2 )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 ) ) ).

% take_bit_int_less_self_iff
thf(fact_9258_take__bit__int__eq__self,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ K2 )
          = K2 ) ) ) ).

% take_bit_int_eq_self
thf(fact_9259_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K2 )
        = K2 )
      = ( ( ord_less_eq_int @ zero_zero_int @ K2 )
        & ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_9260_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_minus_bit0
thf(fact_9261_take__bit__incr__eq,axiom,
    ! [N: nat,K2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K2 )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K2 @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ) ).

% take_bit_incr_eq
thf(fact_9262_take__bit__int__less__eq,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ ( minus_minus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_9263_take__bit__int__greater__eq,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_int @ K2 @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ).

% take_bit_int_greater_eq
thf(fact_9264_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_9265_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_9266_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K2 )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K2 @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_9267_take__bit__minus__small__eq,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ( ( ord_less_eq_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K2 ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_9268_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K2 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_9269_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_9270_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_9271_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K2 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_9272_pred__numeral__inc,axiom,
    ! [K2: num] :
      ( ( pred_numeral @ ( inc @ K2 ) )
      = ( numeral_numeral_nat @ K2 ) ) ).

% pred_numeral_inc
thf(fact_9273_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_9274_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_9275_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_9276_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_9277_Suc__0__and__eq,axiom,
    ! [N: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Suc_0_and_eq
thf(fact_9278_and__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_Suc_0_eq
thf(fact_9279_num__induct,axiom,
    ! [P: num > $o,X: num] :
      ( ( P @ one )
     => ( ! [X5: num] :
            ( ( P @ X5 )
           => ( P @ ( inc @ X5 ) ) )
       => ( P @ X ) ) ) ).

% num_induct
thf(fact_9280_add__inc,axiom,
    ! [X: num,Y: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).

% add_inc
thf(fact_9281_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_9282_inc_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit1 @ X ) )
      = ( bit0 @ ( inc @ X ) ) ) ).

% inc.simps(3)
thf(fact_9283_inc_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit0 @ X ) )
      = ( bit1 @ X ) ) ).

% inc.simps(2)
thf(fact_9284_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_9285_inc__BitM__eq,axiom,
    ! [N: num] :
      ( ( inc @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% inc_BitM_eq
thf(fact_9286_BitM__inc__eq,axiom,
    ! [N: num] :
      ( ( bitM @ ( inc @ N ) )
      = ( bit1 @ N ) ) ).

% BitM_inc_eq
thf(fact_9287_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% and_nat_def
thf(fact_9288_mult__inc,axiom,
    ! [X: num,Y: num] :
      ( ( times_times_num @ X @ ( inc @ Y ) )
      = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).

% mult_inc
thf(fact_9289_atLeastAtMostPlus1__int__conv,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_eq_int @ M2 @ ( plus_plus_int @ one_one_int @ N ) )
     => ( ( set_or1266510415728281911st_int @ M2 @ ( plus_plus_int @ one_one_int @ N ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M2 @ N ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_9290_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I3: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_9291_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N3: nat] :
          ( if_nat
          @ ( ( M3 = zero_zero_nat )
            | ( N3 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_9292_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_9293_and__int_Opsimps,axiom,
    ! [K2: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L ) )
     => ( ( ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K2 @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K2 @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_9294_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
       => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
              & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).

% and_int.pelims
thf(fact_9295_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N3 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_9296_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K2 @ N ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_9297_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K2 @ N ) ) ).

% signed_take_bit_negative_iff
thf(fact_9298_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_9299_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_9300_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).

% bit_minus_numeral_int(1)
thf(fact_9301_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).

% bit_minus_numeral_int(2)
thf(fact_9302_set__encode__insert,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ N @ A4 )
       => ( ( nat_set_encode @ ( insert_nat @ N @ A4 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).

% set_encode_insert
thf(fact_9303_bit__and__int__iff,axiom,
    ! [K2: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K2 @ L ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K2 @ N )
        & ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).

% bit_and_int_iff
thf(fact_9304_lessThan__Suc,axiom,
    ! [K2: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
      = ( insert_nat @ K2 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).

% lessThan_Suc
thf(fact_9305_atMost__Suc,axiom,
    ! [K2: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K2 ) )
      = ( insert_nat @ ( suc @ K2 ) @ ( set_ord_atMost_nat @ K2 ) ) ) ).

% atMost_Suc
thf(fact_9306_bit__not__int__iff_H,axiom,
    ! [K2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) @ N )
      = ( ~ ( bit_se1146084159140164899it_int @ K2 @ N ) ) ) ).

% bit_not_int_iff'
thf(fact_9307_atLeast0__atMost__Suc,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_9308_Icc__eq__insert__lb__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( set_or1269000886237332187st_nat @ M2 @ N )
        = ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_9309_atLeastAtMostSuc__conv,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) )
        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_9310_atLeastAtMost__insertL,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
        = ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% atLeastAtMost_insertL
thf(fact_9311_lessThan__nat__numeral,axiom,
    ! [K2: num] :
      ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K2 ) )
      = ( insert_nat @ ( pred_numeral @ K2 ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K2 ) ) ) ) ).

% lessThan_nat_numeral
thf(fact_9312_atMost__nat__numeral,axiom,
    ! [K2: num] :
      ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K2 ) )
      = ( insert_nat @ ( numeral_numeral_nat @ K2 ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K2 ) ) ) ) ).

% atMost_nat_numeral
thf(fact_9313_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M2: nat,K2: int] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( bit_se1146084159140164899it_int @ K2 @ N )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M2 @ K2 ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_9314_bit__concat__bit__iff,axiom,
    ! [M2: nat,K2: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M2 @ K2 @ L ) @ N )
      = ( ( ( ord_less_nat @ N @ M2 )
          & ( bit_se1146084159140164899it_int @ K2 @ N ) )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9315_signed__take__bit__eq__concat__bit,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K3: int] : ( bit_concat_bit @ N3 @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N3 ) ) ) ) ) ) ).

% signed_take_bit_eq_concat_bit
thf(fact_9316_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N2: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ N2 @ M4 )
             => ( ( bit_se1146084159140164899it_int @ K2 @ M4 )
                = ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( bit_se1146084159140164899it_int @ K2 @ ( minus_minus_nat @ N2 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ).

% int_bit_bound
thf(fact_9317_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_9318_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K3: int,N3: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% bit_int_def
thf(fact_9319_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
        = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_9320_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [K: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L4 ) )
           => ( ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K @ L4 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% and_int.pinduct
thf(fact_9321_set__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N3: nat,K3: int] :
          ( plus_plus_int @ K3
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K3 @ N3 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% set_bit_eq
thf(fact_9322_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% unset_bit_eq
thf(fact_9323_take__bit__Suc__from__most,axiom,
    ! [N: nat,K2: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K2 )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ).

% take_bit_Suc_from_most
thf(fact_9324_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I2: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I2 @ J2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
             => ( P @ I2 @ J2 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_9325_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L2
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_9326_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = one_one_complex ) ) ).

% cis_multiple_2pi
thf(fact_9327_rat__inverse__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,B5: int] : ( if_Pro3027730157355071871nt_int @ ( A5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A5 ) @ B5 ) @ ( abs_abs_int @ A5 ) ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_inverse_code
thf(fact_9328_or__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K2 @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K2 )
        & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% or_nonnegative_int_iff
thf(fact_9329_or__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K2 @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K2 @ zero_zero_int )
        | ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_9330_quotient__of__number_I3_J,axiom,
    ! [K2: num] :
      ( ( quotient_of @ ( numeral_numeral_rat @ K2 ) )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ).

% quotient_of_number(3)
thf(fact_9331_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_9332_or__minus__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).

% or_minus_numerals(2)
thf(fact_9333_or__minus__numerals_I6_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).

% or_minus_numerals(6)
thf(fact_9334_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9335_quotient__of__number_I5_J,axiom,
    ! [K2: num] :
      ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) ) )
      = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) ).

% quotient_of_number(5)
thf(fact_9336_quotient__of__number_I4_J,axiom,
    ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).

% quotient_of_number(4)
thf(fact_9337_bit__or__int__iff,axiom,
    ! [K2: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K2 @ L ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K2 @ N )
        | ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).

% bit_or_int_iff
thf(fact_9338_divide__rat__def,axiom,
    ( divide_divide_rat
    = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).

% divide_rat_def
thf(fact_9339_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
thf(fact_9340_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_9341_OR__lower,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 @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).

% OR_lower
thf(fact_9342_or__greater__eq,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ K2 @ ( bit_se1409905431419307370or_int @ K2 @ L ) ) ) ).

% or_greater_eq
thf(fact_9343_plus__and__or,axiom,
    ! [X: int,Y: int] :
      ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
      = ( plus_plus_int @ X @ Y ) ) ).

% plus_and_or
thf(fact_9344_not__bit__Suc__0__numeral,axiom,
    ! [N: num] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).

% not_bit_Suc_0_numeral
thf(fact_9345_bit__nat__iff,axiom,
    ! [K2: int,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K2 ) @ N )
      = ( ( ord_less_eq_int @ zero_zero_int @ K2 )
        & ( bit_se1146084159140164899it_int @ K2 @ N ) ) ) ).

% bit_nat_iff
thf(fact_9346_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P5: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A5: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B5: int,D2: int] : ( ord_less_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B5 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_code
thf(fact_9347_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P5: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A5: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B5: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B5 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_eq_code
thf(fact_9348_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
        = zero_zero_real )
      = ( member_real @ X @ ring_1_Ints_real ) ) ).

% sin_times_pi_eq_0
thf(fact_9349_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M3: nat,N3: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% bit_nat_def
thf(fact_9350_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% OR_upper
thf(fact_9351_sin__integer__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = zero_zero_real ) ) ).

% sin_integer_2pi
thf(fact_9352_cos__integer__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = one_one_real ) ) ).

% cos_integer_2pi
thf(fact_9353_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_9354_quotient__of__int,axiom,
    ! [A: int] :
      ( ( quotient_of @ ( of_int @ A ) )
      = ( product_Pair_int_int @ A @ one_one_int ) ) ).

% quotient_of_int
thf(fact_9355_rat__minus__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P2 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B5 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_minus_code
thf(fact_9356_rat__plus__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P2 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B5 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_plus_code
thf(fact_9357_or__minus__numerals_I5_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(5)
thf(fact_9358_normalize__denom__zero,axiom,
    ! [P2: int] :
      ( ( normalize @ ( product_Pair_int_int @ P2 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_9359_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_9360_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_9361_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_9362_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_9363_or__minus__numerals_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M2 @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(4)
thf(fact_9364_or__minus__numerals_I8_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M2 @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(8)
thf(fact_9365_or__minus__numerals_I7_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M2 @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(7)
thf(fact_9366_or__minus__numerals_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M2 @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(3)
thf(fact_9367_or__minus__numerals_I1_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(1)
thf(fact_9368_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_9369_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
      = ( bit0 @ one ) ) ).

% or_not_num_neg.simps(4)
thf(fact_9370_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M2 ) )
      = ( bit0 @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ).

% or_not_num_neg.simps(6)
thf(fact_9371_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M2: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit1 @ M2 ) )
      = ( bit1 @ M2 ) ) ).

% or_not_num_neg.simps(3)
thf(fact_9372_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
      = one ) ).

% or_not_num_neg.simps(7)
thf(fact_9373_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M2 ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ).

% or_not_num_neg.simps(5)
thf(fact_9374_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M2 ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ).

% or_not_num_neg.simps(9)
thf(fact_9375_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% or_nat_def
thf(fact_9376_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M2: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit0 @ M2 ) )
      = ( bit1 @ M2 ) ) ).

% or_not_num_neg.simps(2)
thf(fact_9377_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M2 ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ).

% or_not_num_neg.simps(8)
thf(fact_9378_normalize__crossproduct,axiom,
    ! [Q3: int,S2: int,P2: int,R2: int] :
      ( ( Q3 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P2 @ Q3 ) )
            = ( normalize @ ( product_Pair_int_int @ R2 @ S2 ) ) )
         => ( ( times_times_int @ P2 @ S2 )
            = ( times_times_int @ R2 @ Q3 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9379_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa2: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X @ Xa2 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != one ) ) )
       => ( ( ( X = one )
           => ! [M: num] :
                ( ( Xa2
                  = ( bit0 @ M ) )
               => ( Y
                 != ( bit1 @ M ) ) ) )
         => ( ( ( X = one )
             => ! [M: num] :
                  ( ( Xa2
                    = ( bit1 @ M ) )
                 => ( Y
                   != ( bit1 @ M ) ) ) )
           => ( ( ? [N2: num] :
                    ( X
                    = ( bit0 @ N2 ) )
               => ( ( Xa2 = one )
                 => ( Y
                   != ( bit0 @ one ) ) ) )
             => ( ! [N2: num] :
                    ( ( X
                      = ( bit0 @ N2 ) )
                   => ! [M: num] :
                        ( ( Xa2
                          = ( bit0 @ M ) )
                       => ( Y
                         != ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) )
               => ( ! [N2: num] :
                      ( ( X
                        = ( bit0 @ N2 ) )
                     => ! [M: num] :
                          ( ( Xa2
                            = ( bit1 @ M ) )
                         => ( Y
                           != ( bit0 @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) )
                 => ( ( ? [N2: num] :
                          ( X
                          = ( bit1 @ N2 ) )
                     => ( ( Xa2 = one )
                       => ( Y != one ) ) )
                   => ( ! [N2: num] :
                          ( ( X
                            = ( bit1 @ N2 ) )
                         => ! [M: num] :
                              ( ( Xa2
                                = ( bit0 @ M ) )
                             => ( Y
                               != ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) )
                     => ~ ! [N2: num] :
                            ( ( X
                              = ( bit1 @ N2 ) )
                           => ! [M: num] :
                                ( ( Xa2
                                  = ( bit1 @ M ) )
                               => ( Y
                                 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_9380_or__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% or_Suc_0_eq
thf(fact_9381_Suc__0__or__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% Suc_0_or_eq
thf(fact_9382_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_9383_rat__divide__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P2 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B5 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_divide_code
thf(fact_9384_rat__times__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ( quotient_of @ ( times_times_rat @ P2 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ B5 ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_times_code
thf(fact_9385_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9386_Frct__code__post_I5_J,axiom,
    ! [K2: num] :
      ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K2 ) ) ) ).

% Frct_code_post(5)
thf(fact_9387_xor__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_9388_Suc__0__xor__eq,axiom,
    ! [N: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_9389_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).

% horner_sum_of_bool_2_less
thf(fact_9390_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_9391_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_9392_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_9393_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_9394_Frct__code__post_I3_J,axiom,
    ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
    = one_one_rat ) ).

% Frct_code_post(3)
thf(fact_9395_Frct__code__post_I4_J,axiom,
    ! [K2: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K2 ) @ one_one_int ) )
      = ( numeral_numeral_rat @ K2 ) ) ).

% Frct_code_post(4)
thf(fact_9396_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9397_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9398_Sum__Ico__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_9399_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_9400_push__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K2 ) @ zero_zero_int )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_9401_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] :
      ( ( bit_concat_bit @ N @ zero_zero_int @ L )
      = ( bit_se545348938243370406it_int @ N @ L ) ) ).

% concat_bit_of_zero_1
thf(fact_9402_xor__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K2 @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K2 )
        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_9403_xor__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K2 @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K2 @ zero_zero_int )
       != ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_9404_atLeastLessThan__singleton,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ M2 ) )
      = ( insert_nat @ M2 @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9405_push__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_Suc_0
thf(fact_9406_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N3: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_9407_bit__xor__int__iff,axiom,
    ! [K2: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K2 @ L ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K2 @ N )
       != ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).

% bit_xor_int_iff
thf(fact_9408_push__bit__nat__eq,axiom,
    ! [N: nat,K2: int] :
      ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K2 ) )
      = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K2 ) ) ) ).

% push_bit_nat_eq
thf(fact_9409_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
           => ( P @ M3 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X4 ) ) ) ) ).

% all_nat_less_eq
thf(fact_9410_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
            & ( P @ M3 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X4 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9411_XOR__lower,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 @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).

% XOR_lower
thf(fact_9412_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_9413_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M3: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9414_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M3: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_9415_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_9416_bit__push__bit__iff__int,axiom,
    ! [M2: nat,K2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M2 @ K2 ) @ N )
      = ( ( ord_less_eq_nat @ M2 @ N )
        & ( bit_se1146084159140164899it_int @ K2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9417_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N3: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% xor_nat_def
thf(fact_9418_bit__push__bit__iff__nat,axiom,
    ! [M2: nat,Q3: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M2 @ Q3 ) @ N )
      = ( ( ord_less_eq_nat @ M2 @ N )
        & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9419_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N3: nat,K3: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N3 @ K3 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).

% concat_bit_eq
thf(fact_9420_concat__bit__def,axiom,
    ( bit_concat_bit
    = ( ^ [N3: nat,K3: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N3 @ K3 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).

% concat_bit_def
thf(fact_9421_set__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N3: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% set_bit_int_def
thf(fact_9422_atLeastLessThanSuc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
          = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_9423_prod__Suc__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_Suc_fact
thf(fact_9424_prod__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_fact
thf(fact_9425_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N3: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_int_def
thf(fact_9426_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N3: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_nat_def
thf(fact_9427_atLeastLessThan__nat__numeral,axiom,
    ! [M2: nat,K2: num] :
      ( ( ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K2 ) )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K2 ) )
          = ( insert_nat @ ( pred_numeral @ K2 ) @ ( set_or4665077453230672383an_nat @ M2 @ ( pred_numeral @ K2 ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K2 ) )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K2 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_9428_push__bit__minus__one,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% push_bit_minus_one
thf(fact_9429_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% XOR_upper
thf(fact_9430_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_9431_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_9432_sum__power2,axiom,
    ! [K2: nat] :
      ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) @ one_one_nat ) ) ).

% sum_power2
thf(fact_9433_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: nat > nat,B: nat > nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I2 @ J2 )
         => ( ( ord_less_nat @ J2 @ N )
           => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
     => ( ! [I2: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I2 @ J2 )
           => ( ( ord_less_nat @ J2 @ N )
             => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9434_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( K3
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L2 )
          @ ( if_int
            @ ( L2
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K3 )
            @ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9435_not__negative__int__iff,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% not_negative_int_iff
thf(fact_9436_not__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K2 ) )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% not_nonnegative_int_iff
thf(fact_9437_and__minus__minus__numerals,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_9438_or__minus__minus__numerals,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_9439_bit__not__int__iff,axiom,
    ! [K2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ N )
      = ( ~ ( bit_se1146084159140164899it_int @ K2 @ N ) ) ) ).

% bit_not_int_iff
thf(fact_9440_or__int__def,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L2 ) ) ) ) ) ).

% or_int_def
thf(fact_9441_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_9442_and__not__numerals_I1_J,axiom,
    ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = zero_zero_int ) ).

% and_not_numerals(1)
thf(fact_9443_or__not__numerals_I1_J,axiom,
    ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(1)
thf(fact_9444_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
      = ( set_or1266510415728281911st_int @ L @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_9445_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N3: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_9446_xor__int__def,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L2 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L2 ) ) ) ) ).

% xor_int_def
thf(fact_9447_not__int__div__2,axiom,
    ! [K2: int] :
      ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% not_int_div_2
thf(fact_9448_even__not__iff__int,axiom,
    ! [K2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K2 ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) ) ).

% even_not_iff_int
thf(fact_9449_and__not__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = one_one_int ) ).

% and_not_numerals(2)
thf(fact_9450_and__not__numerals_I4_J,axiom,
    ! [M2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ).

% and_not_numerals(4)
thf(fact_9451_or__not__numerals_I4_J,axiom,
    ! [M2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).

% or_not_numerals(4)
thf(fact_9452_or__not__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).

% or_not_numerals(2)
thf(fact_9453_bit__minus__int__iff,axiom,
    ! [K2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K2 ) @ N )
      = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K2 @ one_one_int ) ) @ N ) ) ).

% bit_minus_int_iff
thf(fact_9454_numeral__or__not__num__eq,axiom,
    ! [M2: num,N: num] :
      ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M2 @ N ) )
      = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% numeral_or_not_num_eq
thf(fact_9455_int__numeral__not__or__num__neg,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ) ).

% int_numeral_not_or_num_neg
thf(fact_9456_int__numeral__or__not__num__neg,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M2 @ N ) ) ) ) ).

% int_numeral_or_not_num_neg
thf(fact_9457_and__not__numerals_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(5)
thf(fact_9458_and__not__numerals_I7_J,axiom,
    ! [M2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ).

% and_not_numerals(7)
thf(fact_9459_or__not__numerals_I3_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).

% or_not_numerals(3)
thf(fact_9460_and__not__numerals_I3_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = zero_zero_int ) ).

% and_not_numerals(3)
thf(fact_9461_or__not__numerals_I7_J,axiom,
    ! [M2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(7)
thf(fact_9462_and__not__numerals_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(9)
thf(fact_9463_and__not__numerals_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(6)
thf(fact_9464_or__not__numerals_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% or_not_numerals(6)
thf(fact_9465_or__not__numerals_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_9466_and__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_9467_or__not__numerals_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_9468_or__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_9469_not__int__rec,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_9470_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_9471_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X6: nat > real] :
        ! [J3: nat] :
        ? [M9: nat] :
        ! [M3: nat] :
          ( ( ord_less_eq_nat @ M9 @ M3 )
         => ! [N3: nat] :
              ( ( ord_less_eq_nat @ M9 @ N3 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M3 ) @ ( X6 @ N3 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9472_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_9473_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_9474_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_9475_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,D: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D )
      = ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_9476_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_9477_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_9478_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9479_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] :
      ( ( re @ ( csqrt @ Z ) )
      = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% csqrt.simps(1)
thf(fact_9480_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
      = ( uminus1351360451143612070nteger @ L ) ) ).

% minus_integer_code(2)
thf(fact_9481_minus__integer__code_I1_J,axiom,
    ! [K2: code_integer] :
      ( ( minus_8373710615458151222nteger @ K2 @ zero_z3403309356797280102nteger )
      = K2 ) ).

% minus_integer_code(1)
thf(fact_9482_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_9483_times__integer__code_I1_J,axiom,
    ! [K2: code_integer] :
      ( ( times_3573771949741848930nteger @ K2 @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_9484_times__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_9485_one__complex_Osimps_I1_J,axiom,
    ( ( re @ one_one_complex )
    = one_one_real ) ).

% one_complex.simps(1)
thf(fact_9486_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,X: complex] :
      ( ( re @ ( real_V2046097035970521341omplex @ R2 @ X ) )
      = ( times_times_real @ R2 @ ( re @ X ) ) ) ).

% scaleR_complex.simps(1)
thf(fact_9487_minus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( minus_minus_complex @ X @ Y ) )
      = ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).

% minus_complex.simps(1)
thf(fact_9488_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
thf(fact_9489_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_9490_cos__n__Re__cis__pow__n,axiom,
    ! [N: nat,A: real] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
      = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).

% cos_n_Re_cis_pow_n
thf(fact_9491_csqrt_Ocode,axiom,
    ( csqrt
    = ( ^ [Z2: complex] :
          ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          @ ( times_times_real
            @ ( if_real
              @ ( ( im @ Z2 )
                = zero_zero_real )
              @ one_one_real
              @ ( sgn_sgn_real @ ( im @ Z2 ) ) )
            @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% csqrt.code
thf(fact_9492_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      ( ( im @ ( csqrt @ Z ) )
      = ( times_times_real
        @ ( if_real
          @ ( ( im @ Z )
            = zero_zero_real )
          @ one_one_real
          @ ( sgn_sgn_real @ ( im @ Z ) ) )
        @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt.simps(2)
thf(fact_9493_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
          @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_9494_Im__i__times,axiom,
    ! [Z: complex] :
      ( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
      = ( re @ Z ) ) ).

% Im_i_times
thf(fact_9495_Re__i__times,axiom,
    ! [Z: complex] :
      ( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
      = ( uminus_uminus_real @ ( im @ Z ) ) ) ).

% Re_i_times
thf(fact_9496_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
        | ( ( ( im @ X )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
     => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
        = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).

% csqrt_minus
thf(fact_9497_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
       => ( ( csqrt @ X )
          = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).

% csqrt_of_real_nonpos
thf(fact_9498_imaginary__unit_Osimps_I2_J,axiom,
    ( ( im @ imaginary_unit )
    = one_one_real ) ).

% imaginary_unit.simps(2)
thf(fact_9499_one__complex_Osimps_I2_J,axiom,
    ( ( im @ one_one_complex )
    = zero_zero_real ) ).

% one_complex.simps(2)
thf(fact_9500_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( times_times_int @ Xa2 @ X ) ) ) ).

% times_integer.abs_eq
thf(fact_9501_one__integer__def,axiom,
    ( one_one_Code_integer
    = ( code_integer_of_int @ one_one_int ) ) ).

% one_integer_def
thf(fact_9502_minus__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( minus_minus_int @ Xa2 @ X ) ) ) ).

% minus_integer.abs_eq
thf(fact_9503_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,X: complex] :
      ( ( im @ ( real_V2046097035970521341omplex @ R2 @ X ) )
      = ( times_times_real @ R2 @ ( im @ X ) ) ) ).

% scaleR_complex.simps(2)
thf(fact_9504_minus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( minus_minus_complex @ X @ Y ) )
      = ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).

% minus_complex.simps(2)
thf(fact_9505_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( times_times_complex @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ).

% times_complex.simps(2)
thf(fact_9506_times__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( times_times_complex @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ).

% times_complex.simps(1)
thf(fact_9507_scaleR__complex_Ocode,axiom,
    ( real_V2046097035970521341omplex
    = ( ^ [R5: real,X4: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X4 ) ) @ ( times_times_real @ R5 @ ( im @ X4 ) ) ) ) ) ).

% scaleR_complex.code
thf(fact_9508_minus__complex_Ocode,axiom,
    ( minus_minus_complex
    = ( ^ [X4: complex,Y3: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X4 ) @ ( re @ Y3 ) ) @ ( minus_minus_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) ) ) ).

% minus_complex.code
thf(fact_9509_sin__n__Im__cis__pow__n,axiom,
    ! [N: nat,A: real] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
      = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).

% sin_n_Im_cis_pow_n
thf(fact_9510_Re__exp,axiom,
    ! [Z: complex] :
      ( ( re @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).

% Re_exp
thf(fact_9511_Im__exp,axiom,
    ! [Z: complex] :
      ( ( im @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).

% Im_exp
thf(fact_9512_complex__eq,axiom,
    ! [A: complex] :
      ( A
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).

% complex_eq
thf(fact_9513_times__complex_Ocode,axiom,
    ( times_times_complex
    = ( ^ [X4: complex,Y3: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( im @ Y3 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y3 ) ) ) ) ) ) ).

% times_complex.code
thf(fact_9514_exp__eq__polar,axiom,
    ( exp_complex
    = ( ^ [Z2: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z2 ) ) ) @ ( cis @ ( im @ Z2 ) ) ) ) ) ).

% exp_eq_polar
thf(fact_9515_cmod__power2,axiom,
    ! [Z: complex] :
      ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cmod_power2
thf(fact_9516_Im__power2,axiom,
    ! [X: complex] :
      ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).

% Im_power2
thf(fact_9517_Re__power2,axiom,
    ! [X: complex] :
      ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Re_power2
thf(fact_9518_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
      = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real ) ) ).

% complex_eq_0
thf(fact_9519_norm__complex__def,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z2: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% norm_complex_def
thf(fact_9520_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] :
      ( ( re @ ( invers8013647133539491842omplex @ X ) )
      = ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% inverse_complex.simps(1)
thf(fact_9521_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
      = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_neq_0
thf(fact_9522_Re__divide,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( divide1717551699836669952omplex @ X @ Y ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_divide
thf(fact_9523_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = Z )
     => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
          | ( ( ( re @ W )
              = zero_zero_real )
            & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
       => ( ( csqrt @ Z )
          = W ) ) ) ).

% csqrt_unique
thf(fact_9524_csqrt__square,axiom,
    ! [B: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
        | ( ( ( re @ B )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
     => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = B ) ) ).

% csqrt_square
thf(fact_9525_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] :
      ( ( im @ ( invers8013647133539491842omplex @ X ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% inverse_complex.simps(2)
thf(fact_9526_Im__divide,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( divide1717551699836669952omplex @ X @ Y ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_divide
thf(fact_9527_complex__abs__le__norm,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% complex_abs_le_norm
thf(fact_9528_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real ) ) ).

% complex_unit_circle
thf(fact_9529_inverse__complex_Ocode,axiom,
    ( invers8013647133539491842omplex
    = ( ^ [X4: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X4 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X4 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% inverse_complex.code
thf(fact_9530_Complex__divide,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X4: complex,Y3: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( re @ X4 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% Complex_divide
thf(fact_9531_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_Reals_divide
thf(fact_9532_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_Reals_divide
thf(fact_9533_real__eq__imaginary__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( X
            = ( times_times_complex @ imaginary_unit @ Y ) )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
thf(fact_9534_imaginary__eq__real__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( ( times_times_complex @ imaginary_unit @ Y )
            = X )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
thf(fact_9535_complex__diff__cnj,axiom,
    ! [Z: complex] :
      ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).

% complex_diff_cnj
thf(fact_9536_complex__mult__cnj,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( cnj @ Z ) )
      = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_mult_cnj
thf(fact_9537_complex__cnj__mult,axiom,
    ! [X: complex,Y: complex] :
      ( ( cnj @ ( times_times_complex @ X @ Y ) )
      = ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).

% complex_cnj_mult
thf(fact_9538_complex__cnj__one,axiom,
    ( ( cnj @ one_one_complex )
    = one_one_complex ) ).

% complex_cnj_one
thf(fact_9539_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( ( cnj @ Z )
        = one_one_complex )
      = ( Z = one_one_complex ) ) ).

% complex_cnj_one_iff
thf(fact_9540_complex__cnj__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( cnj @ ( minus_minus_complex @ X @ Y ) )
      = ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).

% complex_cnj_diff
thf(fact_9541_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] :
      ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
      = zero_zero_real ) ).

% complex_In_mult_cnj_zero
thf(fact_9542_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
        = zero_zero_real )
      = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
        = zero_zero_real ) ) ).

% Re_complex_div_eq_0
thf(fact_9543_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
        = zero_zero_real )
      = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
        = zero_zero_real ) ) ).

% Im_complex_div_eq_0
thf(fact_9544_complex__mod__sqrt__Re__mult__cnj,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z2: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) ) ) ) ) ).

% complex_mod_sqrt_Re_mult_cnj
thf(fact_9545_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Re_complex_div_gt_0
thf(fact_9546_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_lt_0
thf(fact_9547_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_le_0
thf(fact_9548_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Re_complex_div_ge_0
thf(fact_9549_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Im_complex_div_gt_0
thf(fact_9550_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_lt_0
thf(fact_9551_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_le_0
thf(fact_9552_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Im_complex_div_ge_0
thf(fact_9553_complex__mod__mult__cnj,axiom,
    ! [Z: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% complex_mod_mult_cnj
thf(fact_9554_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
      & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).

% complex_div_gt_0
thf(fact_9555_complex__norm__square,axiom,
    ! [Z: complex] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).

% complex_norm_square
thf(fact_9556_complex__add__cnj,axiom,
    ! [Z: complex] :
      ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).

% complex_add_cnj
thf(fact_9557_complex__div__cnj,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A5: complex,B5: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A5 @ ( cnj @ B5 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_div_cnj
thf(fact_9558_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).

% cnj_add_mult_eq_Re
thf(fact_9559_integer__of__num_I3_J,axiom,
    ! [N: num] :
      ( ( code_integer_of_num @ ( bit1 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).

% integer_of_num(3)
thf(fact_9560_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K3: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
          @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_9561_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_9562_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa2 ) )
      = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% times_integer.rep_eq
thf(fact_9563_one__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ one_one_Code_integer )
    = one_one_int ) ).

% one_integer.rep_eq
thf(fact_9564_minus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa2 ) )
      = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% minus_integer.rep_eq
thf(fact_9565_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_9566_integer__of__num_I2_J,axiom,
    ! [N: num] :
      ( ( code_integer_of_num @ ( bit0 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).

% integer_of_num(2)
thf(fact_9567_integer__of__num__triv_I2_J,axiom,
    ( ( code_integer_of_num @ ( bit0 @ one ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% integer_of_num_triv(2)
thf(fact_9568_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_9569_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( B3
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( ( A3
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_9570_card__Collect__less__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
      = N ) ).

% card_Collect_less_nat
thf(fact_9571_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_9572_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_atLeastLessThan
thf(fact_9573_card__Collect__le__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N ) ) )
      = ( suc @ N ) ) ).

% card_Collect_le_nat
thf(fact_9574_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_9575_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).

% card_atLeastLessThan_int
thf(fact_9576_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).

% card_atLeastAtMost_int
thf(fact_9577_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M7 )
                & ( ord_less_nat @ K3 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_9578_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M7 )
                  & ( ord_less_nat @ K3 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_9579_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_9580_subset__card__intvl__is__intvl,axiom,
    ! [A4: set_nat,K2: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A4 ) ) ) )
     => ( A4
        = ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A4 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_9581_nat__of__integer__code__post_I3_J,axiom,
    ! [K2: num] :
      ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K2 ) )
      = ( numeral_numeral_nat @ K2 ) ) ).

% nat_of_integer_code_post(3)
thf(fact_9582_nat__of__integer__code__post_I2_J,axiom,
    ( ( code_nat_of_integer @ one_one_Code_integer )
    = one_one_nat ) ).

% nat_of_integer_code_post(2)
thf(fact_9583_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_9584_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ S3 ) ) ).

% card_sum_le_nat_sum
thf(fact_9585_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z2: complex] :
                  ( ( power_power_complex @ Z2 @ N )
                  = C ) ) )
          = N ) ) ) ).

% card_nth_roots
thf(fact_9586_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) ) )
        = N ) ) ).

% card_roots_unity_eq
thf(fact_9587_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( A3
                   => ( Y
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A3
                   => ( ( B3
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_9588_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv ) )
               => ( ~ Y
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
           => ( ! [Uu: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu @ $true ) )
                 => ( ~ Y
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) ) )
             => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
                   => ( Y
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                     => ( ~ Y
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_9589_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ! [Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ $true @ Uv ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) )
         => ( ! [Uu: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uu @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_9590_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_9591_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_9592_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_9593_less__eq__nat_Osimps_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M2 ) @ N ) ) ).

% less_eq_nat.simps(2)
thf(fact_9594_max__Suc2,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_max_nat @ M2 @ ( suc @ N ) )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ M5 @ N ) )
        @ M2 ) ) ).

% max_Suc2
thf(fact_9595_max__Suc1,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_max_nat @ ( suc @ N ) @ M2 )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ N @ M5 ) )
        @ M2 ) ) ).

% max_Suc1
thf(fact_9596_diff__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K3: nat] : K3
        @ ( minus_minus_nat @ M2 @ N ) ) ) ).

% diff_Suc
thf(fact_9597_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_9598_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X23: nat] : X23 ) ) ).

% pred_def
thf(fact_9599_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_9600_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K3 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K3 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M3 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_9601_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K2 ) ) ) ) ) ).

% drop_bit_numeral_minus_bit1
thf(fact_9602_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9603_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K2 ) @ zero_zero_int )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9604_drop__bit__minus__one,axiom,
    ! [N: nat] :
      ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% drop_bit_minus_one
thf(fact_9605_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_9606_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).

% drop_bit_numeral_minus_bit0
thf(fact_9607_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K2 ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_9608_drop__bit__push__bit__int,axiom,
    ! [M2: nat,N: nat,K2: int] :
      ( ( bit_se8568078237143864401it_int @ M2 @ ( bit_se545348938243370406it_int @ N @ K2 ) )
      = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M2 @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M2 ) @ K2 ) ) ) ).

% drop_bit_push_bit_int
thf(fact_9609_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N3: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% drop_bit_int_def
thf(fact_9610_Suc__0__div__numeral,axiom,
    ! [K2: num] :
      ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K2 ) )
      = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K2 ) ) ) ).

% Suc_0_div_numeral
thf(fact_9611_Suc__0__mod__numeral,axiom,
    ! [K2: num] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K2 ) )
      = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K2 ) ) ) ).

% Suc_0_mod_numeral
thf(fact_9612_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S5: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S5 ) ) @ ( S5 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_9613_drop__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_9614_fst__divmod__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( product_fst_nat_nat @ ( divmod_nat @ M2 @ N ) )
      = ( divide_divide_nat @ M2 @ N ) ) ).

% fst_divmod_nat
thf(fact_9615_drop__bit__nat__eq,axiom,
    ! [N: nat,K2: int] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K2 ) )
      = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K2 ) ) ) ).

% drop_bit_nat_eq
thf(fact_9616_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N3: nat,M3: nat] : ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9617_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_9618_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
              @ ( produc6916734918728496179nteger
                @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S5 ) ) )
                @ ( code_divmod_abs @ K3 @ L2 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S5 ) ) )
                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_9619_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_9620_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_9621_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X4: nat,Y3: nat] : ( if_Pro3027730157355071871nt_int @ ( Y3 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Y3 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_9622_rat__sgn__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P2 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P2 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_9623_one__mod__minus__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).

% one_mod_minus_numeral
thf(fact_9624_minus__one__mod__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_mod_numeral
thf(fact_9625_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_9626_Divides_Oadjust__mod__def,axiom,
    ( adjust_mod
    = ( ^ [L2: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L2 @ R5 ) ) ) ) ).

% Divides.adjust_mod_def
thf(fact_9627_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_9628_normalize__def,axiom,
    ( normalize
    = ( ^ [P5: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P5 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P5 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_9629_gcd__1__int,axiom,
    ! [M2: int] :
      ( ( gcd_gcd_int @ M2 @ one_one_int )
      = one_one_int ) ).

% gcd_1_int
thf(fact_9630_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U3: int,V2: int] :
      ( ( plus_plus_int @ ( times_times_int @ U3 @ X ) @ ( times_times_int @ V2 @ Y ) )
      = ( gcd_gcd_int @ X @ Y ) ) ).

% bezout_int
thf(fact_9631_gcd__mult__distrib__int,axiom,
    ! [K2: int,M2: int,N: int] :
      ( ( times_times_int @ ( abs_abs_int @ K2 ) @ ( gcd_gcd_int @ M2 @ N ) )
      = ( gcd_gcd_int @ ( times_times_int @ K2 @ M2 ) @ ( times_times_int @ K2 @ N ) ) ) ).

% gcd_mult_distrib_int
thf(fact_9632_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M2: nat] :
      ( ! [K: nat] :
          ( ( ord_less_nat @ N @ K )
         => ( P @ K ) )
     => ( ! [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K ) ) )
       => ( P @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_9633_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).

% card_greaterThanLessThan_int
thf(fact_9634_gcd__1__nat,axiom,
    ! [M2: nat] :
      ( ( gcd_gcd_nat @ M2 @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_9635_gcd__Suc__0,axiom,
    ! [M2: nat] :
      ( ( gcd_gcd_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9636_gcd__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M2 @ N ) )
      = ( ( M2 != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9637_gcd__mult__distrib__nat,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( times_times_nat @ K2 @ ( gcd_gcd_nat @ M2 @ N ) )
      = ( gcd_gcd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) ) ) ).

% gcd_mult_distrib_nat
thf(fact_9638_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_9639_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_9640_gcd__diff1__nat,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M2 @ N ) @ N )
        = ( gcd_gcd_nat @ M2 @ N ) ) ) ).

% gcd_diff1_nat
thf(fact_9641_gcd__diff2__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M2 ) @ N )
        = ( gcd_gcd_nat @ M2 @ N ) ) ) ).

% gcd_diff2_nat
thf(fact_9642_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X5: nat,Y4: nat] :
          ( ( times_times_nat @ A @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_9643_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X5: nat,Y4: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B @ Y4 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9644_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
      = ( set_or5832277885323065728an_int @ L @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9645_bezw__aux,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
      = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).

% bezw_aux
thf(fact_9646_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_9647_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M3: nat] :
          ? [N3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N3 )
            & ( member_nat @ N3 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_9648_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_9649_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_9650_tanh__real__bounds,axiom,
    ! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).

% tanh_real_bounds
thf(fact_9651_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M3: nat] :
          ? [N3: nat] :
            ( ( ord_less_nat @ M3 @ N3 )
            & ( member_nat @ N3 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_9652_unbounded__k__infinite,axiom,
    ! [K2: nat,S3: set_nat] :
      ( ! [M: nat] :
          ( ( ord_less_nat @ K2 @ M )
         => ? [N9: nat] :
              ( ( ord_less_nat @ M @ N9 )
              & ( member_nat @ N9 @ S3 ) ) )
     => ~ ( finite_finite_nat @ S3 ) ) ).

% unbounded_k_infinite
thf(fact_9653_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ? [R4: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R4 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
          & ! [N9: nat] :
              ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S3 ) )
             => ( member_nat @ ( R4 @ N9 ) @ S3 ) ) ) ) ).

% finite_enumerate
thf(fact_9654_xor__minus__numerals_I1_J,axiom,
    ! [N: num,K2: int] :
      ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K2 )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K2 ) ) ) ).

% xor_minus_numerals(1)
thf(fact_9655_xor__minus__numerals_I2_J,axiom,
    ! [K2: int,N: num] :
      ( ( bit_se6526347334894502574or_int @ K2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K2 @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).

% xor_minus_numerals(2)
thf(fact_9656_sub__BitM__One__eq,axiom,
    ! [N: num] :
      ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).

% sub_BitM_One_eq
thf(fact_9657_Suc__funpow,axiom,
    ! [N: nat] :
      ( ( compow_nat_nat @ N @ suc )
      = ( plus_plus_nat @ N ) ) ).

% Suc_funpow
thf(fact_9658_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K3 )
                  = ( sgn_sgn_Code_integer @ L2 ) )
                @ ( code_divmod_abs @ K3 @ L2 )
                @ ( produc6916734918728496179nteger
                  @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S5 ) ) )
                  @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_9659_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X4: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X4 )
    @ ^ [X4: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X4 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_9660_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_9661_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X4: nat,Y3: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) )
          @ Xa2
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_9662_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( gcd_Gcd_nat @ M7 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_9663_Gcd__nat__eq__one,axiom,
    ! [N5: set_nat] :
      ( ( member_nat @ one_one_nat @ N5 )
     => ( ( gcd_Gcd_nat @ N5 )
        = one_one_nat ) ) ).

% Gcd_nat_eq_one
thf(fact_9664_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X5: nat,Y4: nat] :
          ( Z
         != ( abs_Integ @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) ) ).

% eq_Abs_Integ
thf(fact_9665_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( nat2 @ ( abs_Integ @ X ) )
      = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).

% nat.abs_eq
thf(fact_9666_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_9667_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_9668_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X4: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X4 )
          @ X ) ) ) ).

% uminus_int.abs_eq
thf(fact_9669_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9670_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
        @ ^ [X4: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) )
        @ Xa2
        @ X ) ) ).

% less_int.abs_eq
thf(fact_9671_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X4: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) )
        @ Xa2
        @ X ) ) ).

% less_eq_int.abs_eq
thf(fact_9672_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X4: nat,Y3: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y3 @ V4 ) ) )
          @ Xa2
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_9673_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X4: nat,Y3: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) )
          @ Xa2
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_9674_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( suc @ N ) )
          = ( inc @ ( num_of_nat @ N ) ) ) )
      & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( suc @ N ) )
          = one ) ) ) ).

% num_of_nat.simps(2)
thf(fact_9675_num__of__nat__numeral__eq,axiom,
    ! [Q3: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q3 ) )
      = Q3 ) ).

% num_of_nat_numeral_eq
thf(fact_9676_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9677_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
        = N ) ) ).

% numeral_num_of_nat
thf(fact_9678_num__of__nat__One,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ one_one_nat )
     => ( ( num_of_nat @ N )
        = one ) ) ).

% num_of_nat_One
thf(fact_9679_num__of__nat__double,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
        = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).

% num_of_nat_double
thf(fact_9680_num__of__nat__plus__distrib,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( plus_plus_nat @ M2 @ N ) )
          = ( plus_plus_num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_9681_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y3: nat,Z2: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y3 @ V4 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
          @ ( rep_Integ @ X4 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9682_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y3: nat,Z2: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y3 @ V4 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
          @ ( rep_Integ @ X4 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9683_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M3: nat,N3: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M3 @ N3 ) ) @ M3 ) ) ) ).

% prod_encode_def
thf(fact_9684_prod__encode__eq,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_encode @ X )
        = ( nat_prod_encode @ Y ) )
      = ( X = Y ) ) ).

% prod_encode_eq
thf(fact_9685_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_9686_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_9687_nat_Orep__eq,axiom,
    ( nat2
    = ( ^ [X4: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X4 ) ) ) ) ).

% nat.rep_eq
thf(fact_9688_prod__encode__prod__decode__aux,axiom,
    ! [K2: nat,M2: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K2 @ M2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ K2 ) @ M2 ) ) ).

% prod_encode_prod_decode_aux
thf(fact_9689_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X4: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X4 ) ) ) ) ).

% uminus_int_def
thf(fact_9690_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9691_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y3: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_9692_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y3: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_9693_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y3: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y3 @ V4 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_9694_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9695_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit1 @ Y ) )
      = ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).

% pow.simps(3)
thf(fact_9696_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_greaterThanAtMost
thf(fact_9697_sqr_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).

% sqr.simps(2)
thf(fact_9698_sqr_Osimps_I1_J,axiom,
    ( ( sqr @ one )
    = one ) ).

% sqr.simps(1)
thf(fact_9699_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_9700_sqr__conv__mult,axiom,
    ( sqr
    = ( ^ [X4: num] : ( times_times_num @ X4 @ X4 ) ) ) ).

% sqr_conv_mult
thf(fact_9701_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit0 @ Y ) )
      = ( sqr @ ( pow @ X @ Y ) ) ) ).

% pow.simps(2)
thf(fact_9702_sqr_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).

% sqr.simps(3)
thf(fact_9703_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( Y
            = ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                        @ ( 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 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X4: nat] :
                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_9704_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).

% card_greaterThanAtMost_int
thf(fact_9705_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
      = ( set_or6656581121297822940st_int @ L @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_9706_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
      = ( ( Deg = Deg4 )
        & ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
           => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
            & ! [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
          @ ( 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 ) )
                & ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                    & ! [X4: nat] :
                        ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X4 )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_9707_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( Deg2 = Xa2 )
                & ! [X5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                   => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                & ( case_o184042715313410164at_nat
                  @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                    & ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                  @ ( 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 ) )
                        & ! [I3: nat] :
                            ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                            & ! [X4: nat] :
                                ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                 => ( ( ord_less_nat @ Mi3 @ X4 )
                                    & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_9708_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ( 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 @ TreeList2 )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( case_o184042715313410164at_nat
                    @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                    @ ( 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 ) )
                          & ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X4: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                              & ! [X4: nat] :
                                  ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                   => ( ( ord_less_nat @ Mi3 @ X4 )
                                      & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_9709_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 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) )
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
                 => ( ( Deg2 = Xa2 )
                    & ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( case_o184042715313410164at_nat
                      @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                        & ! [X4: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                           => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                      @ ( 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 ) )
                            & ! [I3: nat] :
                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X4: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                & ! [X4: nat] :
                                    ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                     => ( ( ord_less_nat @ Mi3 @ X4 )
                                        & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_9710_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 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) )
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( 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 @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                        @ ( 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 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X4: nat] :
                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_9711_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 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( Y
                  = ( Xa2 = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                        @ ( 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 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X4: nat] :
                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_9712_take__bit__numeral__minus__numeral__int,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M2 ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_int @ Q4 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M2 ) @ N ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_9713_and__minus__numerals_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_9714_take__bit__num__simps_I1_J,axiom,
    ! [M2: num] :
      ( ( bit_take_bit_num @ zero_zero_nat @ M2 )
      = none_num ) ).

% take_bit_num_simps(1)
thf(fact_9715_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(2)
thf(fact_9716_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(5)
thf(fact_9717_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M2: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M2 ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ N @ M2 ) ) ) ).

% take_bit_num_simps(3)
thf(fact_9718_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M2: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M2 ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_9719_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M2: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M2 ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M2 ) ) ) ).

% take_bit_num_simps(6)
thf(fact_9720_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M2: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M2 ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_9721_and__minus__numerals_I8_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_9722_and__minus__numerals_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_9723_and__minus__numerals_I7_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_9724_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one @ one )
    = none_num ) ).

% and_not_num.simps(1)
thf(fact_9725_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M2: num] :
      ( ( bit_take_bit_num @ N @ ( bit0 @ M2 ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N3: nat] :
            ( case_o6005452278849405969um_num @ none_num
            @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
            @ ( bit_take_bit_num @ N3 @ M2 ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9726_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ N @ one )
      = ( case_nat_option_num @ none_num
        @ ^ [N3: nat] : ( some_num @ one )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_9727_and__not__num_Osimps_I4_J,axiom,
    ! [M2: num] :
      ( ( bit_and_not_num @ ( bit0 @ M2 ) @ one )
      = ( some_num @ ( bit0 @ M2 ) ) ) ).

% and_not_num.simps(4)
thf(fact_9728_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
      = ( some_num @ one ) ) ).

% and_not_num.simps(2)
thf(fact_9729_GreatestI__nat,axiom,
    ! [P: nat > $o,K2: nat,B: nat] :
      ( ( P @ K2 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_9730_Greatest__le__nat,axiom,
    ! [P: nat > $o,K2: nat,B: nat] :
      ( ( P @ K2 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_9731_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_9732_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
      = none_num ) ).

% and_not_num.simps(3)
thf(fact_9733_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M2: num] :
      ( ( bit_take_bit_num @ N @ ( bit1 @ M2 ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N3: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N3 @ M2 ) ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_9734_and__not__num_Osimps_I7_J,axiom,
    ! [M2: num] :
      ( ( bit_and_not_num @ ( bit1 @ M2 ) @ one )
      = ( some_num @ ( bit0 @ M2 ) ) ) ).

% and_not_num.simps(7)
thf(fact_9735_and__not__num__eq__Some__iff,axiom,
    ! [M2: num,N: num,Q3: num] :
      ( ( ( bit_and_not_num @ M2 @ N )
        = ( some_num @ Q3 ) )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = ( numeral_numeral_int @ Q3 ) ) ) ).

% and_not_num_eq_Some_iff
thf(fact_9736_and__not__num_Osimps_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_and_not_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
        @ ( bit_and_not_num @ M2 @ N ) ) ) ).

% and_not_num.simps(8)
thf(fact_9737_and__not__num__eq__None__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( bit_and_not_num @ M2 @ N )
        = none_num )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
thf(fact_9738_int__numeral__not__and__num,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M2 ) ) ) ).

% int_numeral_not_and_num
thf(fact_9739_int__numeral__and__not__num,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ N ) ) ) ).

% int_numeral_and_not_num
thf(fact_9740_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N3: nat,M3: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M3 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9741_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N3: nat,M3: num] :
          ( produc478579273971653890on_num
          @ ^ [A5: nat,X4: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P5: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
                      @ ( bit_take_bit_num @ O @ P5 ) )
                  @ ^ [P5: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
                  @ X4 )
              @ A5 )
          @ ( product_Pair_nat_num @ N3 @ M3 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_9742_and__not__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( ( bit_and_not_num @ X @ Xa2 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ( ? [N2: num] :
                  ( Xa2
                  = ( bit0 @ N2 ) )
             => ( Y
               != ( some_num @ one ) ) ) )
         => ( ( ( X = one )
             => ( ? [N2: num] :
                    ( Xa2
                    = ( bit1 @ N2 ) )
               => ( Y != none_num ) ) )
           => ( ! [M: num] :
                  ( ( X
                    = ( bit0 @ M ) )
                 => ( ( Xa2 = one )
                   => ( Y
                     != ( some_num @ ( bit0 @ M ) ) ) ) )
             => ( ! [M: num] :
                    ( ( X
                      = ( bit0 @ M ) )
                   => ! [N2: num] :
                        ( ( Xa2
                          = ( bit0 @ N2 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ) )
               => ( ! [M: num] :
                      ( ( X
                        = ( bit0 @ M ) )
                     => ! [N2: num] :
                          ( ( Xa2
                            = ( bit1 @ N2 ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ) )
                 => ( ! [M: num] :
                        ( ( X
                          = ( bit1 @ M ) )
                       => ( ( Xa2 = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M ) ) ) ) )
                   => ( ! [M: num] :
                          ( ( X
                            = ( bit1 @ M ) )
                         => ! [N2: num] :
                              ( ( Xa2
                                = ( bit0 @ N2 ) )
                             => ( Y
                               != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                  @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                  @ ( bit_and_not_num @ M @ N2 ) ) ) ) )
                     => ~ ! [M: num] :
                            ( ( X
                              = ( bit1 @ M ) )
                           => ! [N2: num] :
                                ( ( Xa2
                                  = ( bit1 @ N2 ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
thf(fact_9743_and__not__num_Osimps_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_and_not_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N ) ) ) ).

% and_not_num.simps(5)
thf(fact_9744_and__not__num_Osimps_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_and_not_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N ) ) ) ).

% and_not_num.simps(6)
thf(fact_9745_and__not__num_Osimps_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_and_not_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N ) ) ) ).

% and_not_num.simps(9)
thf(fact_9746_and__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X @ Xa2 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y
             != ( some_num @ one ) ) ) )
       => ( ( ( X = one )
           => ( ? [N2: num] :
                  ( Xa2
                  = ( bit0 @ N2 ) )
             => ( Y != none_num ) ) )
         => ( ( ( X = one )
             => ( ? [N2: num] :
                    ( Xa2
                    = ( bit1 @ N2 ) )
               => ( Y
                 != ( some_num @ one ) ) ) )
           => ( ( ? [M: num] :
                    ( X
                    = ( bit0 @ M ) )
               => ( ( Xa2 = one )
                 => ( Y != none_num ) ) )
             => ( ! [M: num] :
                    ( ( X
                      = ( bit0 @ M ) )
                   => ! [N2: num] :
                        ( ( Xa2
                          = ( bit0 @ N2 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) )
               => ( ! [M: num] :
                      ( ( X
                        = ( bit0 @ M ) )
                     => ! [N2: num] :
                          ( ( Xa2
                            = ( bit1 @ N2 ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) )
                 => ( ( ? [M: num] :
                          ( X
                          = ( bit1 @ M ) )
                     => ( ( Xa2 = one )
                       => ( Y
                         != ( some_num @ one ) ) ) )
                   => ( ! [M: num] :
                          ( ( X
                            = ( bit1 @ M ) )
                         => ! [N2: num] :
                              ( ( Xa2
                                = ( bit0 @ N2 ) )
                             => ( Y
                               != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) )
                     => ~ ! [M: num] :
                            ( ( X
                              = ( bit1 @ M ) )
                           => ! [N2: num] :
                                ( ( Xa2
                                  = ( bit1 @ N2 ) )
                               => ( Y
                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                    @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
thf(fact_9747_xor__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X @ Xa2 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ! [N2: num] :
                ( ( Xa2
                  = ( bit0 @ N2 ) )
               => ( Y
                 != ( some_num @ ( bit1 @ N2 ) ) ) ) )
         => ( ( ( X = one )
             => ! [N2: num] :
                  ( ( Xa2
                    = ( bit1 @ N2 ) )
                 => ( Y
                   != ( some_num @ ( bit0 @ N2 ) ) ) ) )
           => ( ! [M: num] :
                  ( ( X
                    = ( bit0 @ M ) )
                 => ( ( Xa2 = one )
                   => ( Y
                     != ( some_num @ ( bit1 @ M ) ) ) ) )
             => ( ! [M: num] :
                    ( ( X
                      = ( bit0 @ M ) )
                   => ! [N2: num] :
                        ( ( Xa2
                          = ( bit0 @ N2 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) )
               => ( ! [M: num] :
                      ( ( X
                        = ( bit0 @ M ) )
                     => ! [N2: num] :
                          ( ( Xa2
                            = ( bit1 @ N2 ) )
                         => ( Y
                           != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ) )
                 => ( ! [M: num] :
                        ( ( X
                          = ( bit1 @ M ) )
                       => ( ( Xa2 = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M ) ) ) ) )
                   => ( ! [M: num] :
                          ( ( X
                            = ( bit1 @ M ) )
                         => ! [N2: num] :
                              ( ( Xa2
                                = ( bit0 @ N2 ) )
                             => ( Y
                               != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ) )
                     => ~ ! [M: num] :
                            ( ( X
                              = ( bit1 @ M ) )
                           => ! [N2: num] :
                                ( ( Xa2
                                  = ( bit1 @ N2 ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
thf(fact_9748_and__num_Osimps_I1_J,axiom,
    ( ( bit_un7362597486090784418nd_num @ one @ one )
    = ( some_num @ one ) ) ).

% and_num.simps(1)
thf(fact_9749_xor__num_Osimps_I1_J,axiom,
    ( ( bit_un2480387367778600638or_num @ one @ one )
    = none_num ) ).

% xor_num.simps(1)
thf(fact_9750_xor__num_Osimps_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ).

% xor_num.simps(5)
thf(fact_9751_and__num_Osimps_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).

% and_num.simps(5)
thf(fact_9752_and__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
      = ( some_num @ one ) ) ).

% and_num.simps(3)
thf(fact_9753_and__num_Osimps_I7_J,axiom,
    ! [M2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M2 ) @ one )
      = ( some_num @ one ) ) ).

% and_num.simps(7)
thf(fact_9754_and__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
      = none_num ) ).

% and_num.simps(2)
thf(fact_9755_and__num_Osimps_I4_J,axiom,
    ! [M2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M2 ) @ one )
      = none_num ) ).

% and_num.simps(4)
thf(fact_9756_and__num_Osimps_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).

% and_num.simps(8)
thf(fact_9757_and__num_Osimps_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).

% and_num.simps(6)
thf(fact_9758_xor__num_Osimps_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ).

% xor_num.simps(9)
thf(fact_9759_xor__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
      = ( some_num @ ( bit1 @ N ) ) ) ).

% xor_num.simps(2)
thf(fact_9760_xor__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
      = ( some_num @ ( bit0 @ N ) ) ) ).

% xor_num.simps(3)
thf(fact_9761_xor__num_Osimps_I4_J,axiom,
    ! [M2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M2 ) @ one )
      = ( some_num @ ( bit1 @ M2 ) ) ) ).

% xor_num.simps(4)
thf(fact_9762_xor__num_Osimps_I7_J,axiom,
    ! [M2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M2 ) @ one )
      = ( some_num @ ( bit0 @ M2 ) ) ) ).

% xor_num.simps(7)
thf(fact_9763_and__num_Osimps_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
        @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).

% and_num.simps(9)
thf(fact_9764_xor__num_Osimps_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).

% xor_num.simps(8)
thf(fact_9765_xor__num_Osimps_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).

% xor_num.simps(6)
thf(fact_9766_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
thf(fact_9767_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
thf(fact_9768_Rats__eq__int__div__nat,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I3: int,N3: nat] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
          & ( N3 != zero_zero_nat ) ) ) ) ).

% Rats_eq_int_div_nat
thf(fact_9769_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
      & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).

% rat_floor_lemma
thf(fact_9770_Rats__abs__iff,axiom,
    ! [X: real] :
      ( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
      = ( member_real @ X @ field_5140801741446780682s_real ) ) ).

% Rats_abs_iff
thf(fact_9771_mult__rat,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
      = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).

% mult_rat
thf(fact_9772_divide__rat,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
      = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).

% divide_rat
thf(fact_9773_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% less_rat
thf(fact_9774_add__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% add_rat
thf(fact_9775_le__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% le_rat
thf(fact_9776_diff__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% diff_rat
thf(fact_9777_sgn__rat,axiom,
    ! [A: int,B: int] :
      ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
      = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).

% sgn_rat
thf(fact_9778_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X5: real] :
      ( ( member_real @ X5 @ field_5140801741446780682s_real )
      & ( ord_less_real @ X5 @ X ) ) ).

% Rats_no_bot_less
thf(fact_9779_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [X5: real] :
          ( ( member_real @ X5 @ field_5140801741446780682s_real )
          & ( ord_less_real @ X @ X5 )
          & ( ord_less_real @ X5 @ Y ) ) ) ).

% Rats_dense_in_real
thf(fact_9780_Rats__no__top__le,axiom,
    ! [X: real] :
    ? [X5: real] :
      ( ( member_real @ X5 @ field_5140801741446780682s_real )
      & ( ord_less_eq_real @ X @ X5 ) ) ).

% Rats_no_top_le
thf(fact_9781_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ( fract @ A @ B )
            = ( fract @ C @ D ) )
          = ( ( times_times_int @ A @ D )
            = ( times_times_int @ C @ B ) ) ) ) ) ).

% eq_rat(1)
thf(fact_9782_mult__rat__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( fract @ A @ B ) ) ) ).

% mult_rat_cancel
thf(fact_9783_eq__rat_I2_J,axiom,
    ! [A: int] :
      ( ( fract @ A @ zero_zero_int )
      = ( fract @ zero_zero_int @ one_one_int ) ) ).

% eq_rat(2)
thf(fact_9784_Fract__of__nat__eq,axiom,
    ! [K2: nat] :
      ( ( fract @ ( semiri1314217659103216013at_int @ K2 ) @ one_one_int )
      = ( semiri681578069525770553at_rat @ K2 ) ) ).

% Fract_of_nat_eq
thf(fact_9785_One__rat__def,axiom,
    ( one_one_rat
    = ( fract @ one_one_int @ one_one_int ) ) ).

% One_rat_def
thf(fact_9786_Fract__of__int__eq,axiom,
    ! [K2: int] :
      ( ( fract @ K2 @ one_one_int )
      = ( ring_1_of_int_rat @ K2 ) ) ).

% Fract_of_int_eq
thf(fact_9787_Zero__rat__def,axiom,
    ( zero_zero_rat
    = ( fract @ zero_zero_int @ one_one_int ) ) ).

% Zero_rat_def
thf(fact_9788_rat__number__expand_I3_J,axiom,
    ( numeral_numeral_rat
    = ( ^ [K3: num] : ( fract @ ( numeral_numeral_int @ K3 ) @ one_one_int ) ) ) ).

% rat_number_expand(3)
thf(fact_9789_rat__number__collapse_I3_J,axiom,
    ! [W: num] :
      ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
      = ( numeral_numeral_rat @ W ) ) ).

% rat_number_collapse(3)
thf(fact_9790_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% one_less_Fract_iff
thf(fact_9791_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
        = ( ord_less_int @ A @ B ) ) ) ).

% Fract_less_one_iff
thf(fact_9792_Rats__eq__int__div__int,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I3: int,J3: int] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( ring_1_of_int_real @ J3 ) ) )
          & ( J3 != zero_zero_int ) ) ) ) ).

% Rats_eq_int_div_int
thf(fact_9793_rat__number__collapse_I5_J,axiom,
    ( ( fract @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% rat_number_collapse(5)
thf(fact_9794_Fract__add__one,axiom,
    ! [N: int,M2: int] :
      ( ( N != zero_zero_int )
     => ( ( fract @ ( plus_plus_int @ M2 @ N ) @ N )
        = ( plus_plus_rat @ ( fract @ M2 @ N ) @ one_one_rat ) ) ) ).

% Fract_add_one
thf(fact_9795_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% one_le_Fract_iff
thf(fact_9796_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% Fract_le_one_iff
thf(fact_9797_rat__number__expand_I5_J,axiom,
    ! [K2: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) )
      = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) ).

% rat_number_expand(5)
thf(fact_9798_rat__number__collapse_I4_J,axiom,
    ! [W: num] :
      ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).

% rat_number_collapse(4)
thf(fact_9799_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ 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
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9800_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_9801_bij__betw__Suc,axiom,
    ! [M7: set_nat,N5: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N5 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N5 ) ) ).

% bij_betw_Suc
thf(fact_9802_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_9803_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_9804_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A4 ) ) ).

% zero_notin_Suc_image
thf(fact_9805_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X4: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ) ).

% less_int_def
thf(fact_9806_less__eq__int__def,axiom,
    ( ord_less_eq_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X4: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_9807_nat__def,axiom,
    ( nat2
    = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).

% nat_def
thf(fact_9808_image__Suc__lessThan,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).

% image_Suc_lessThan
thf(fact_9809_image__Suc__atMost,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).

% image_Suc_atMost
thf(fact_9810_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9811_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9812_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_9813_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_9814_positive__rat,axiom,
    ! [A: int,B: int] :
      ( ( positive @ ( fract @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% positive_rat
thf(fact_9815_Rat_Opositive__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( positive @ X )
     => ( ( positive @ Y )
       => ( positive @ ( times_times_rat @ X @ Y ) ) ) ) ).

% Rat.positive_mult
thf(fact_9816_less__rat__def,axiom,
    ( ord_less_rat
    = ( ^ [X4: rat,Y3: rat] : ( positive @ ( minus_minus_rat @ Y3 @ X4 ) ) ) ) ).

% less_rat_def
thf(fact_9817_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X4: int] : ( plus_plus_int @ X4 @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9818_Rat_Opositive_Orep__eq,axiom,
    ( positive
    = ( ^ [X4: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X4 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X4 ) ) ) ) ) ) ).

% Rat.positive.rep_eq
thf(fact_9819_Rat_Opositive__def,axiom,
    ( positive
    = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
      @ ^ [X4: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ).

% Rat.positive_def
thf(fact_9820_of__rat__dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Q2: rat] :
          ( ( ord_less_real @ X @ ( field_7254667332652039916t_real @ Q2 ) )
          & ( ord_less_real @ ( field_7254667332652039916t_real @ Q2 ) @ Y ) ) ) ).

% of_rat_dense
thf(fact_9821_plus__rat__def,axiom,
    ( plus_plus_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X4: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).

% plus_rat_def
thf(fact_9822_inverse__rat__def,axiom,
    ( inverse_inverse_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X4: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int
          @ ( ( product_fst_int_int @ X4 )
            = zero_zero_int )
          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
          @ ( product_Pair_int_int @ ( product_snd_int_int @ X4 ) @ ( product_fst_int_int @ X4 ) ) ) ) ) ).

% inverse_rat_def
thf(fact_9823_one__rat__def,axiom,
    ( one_one_rat
    = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).

% one_rat_def
thf(fact_9824_Fract_Oabs__eq,axiom,
    ( fract
    = ( ^ [Xa4: int,X4: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X4 ) ) ) ) ) ).

% Fract.abs_eq
thf(fact_9825_zero__rat__def,axiom,
    ( zero_zero_rat
    = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).

% zero_rat_def
thf(fact_9826_times__rat__def,axiom,
    ( times_times_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X4: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).

% times_rat_def
thf(fact_9827_plus__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
      ( ( ratrel @ Xa2 @ Xa2 )
     => ( ( ratrel @ X @ X )
       => ( ( plus_plus_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).

% plus_rat.abs_eq
thf(fact_9828_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
        = ( abs_Rat
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_fst_int_int @ X )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).

% inverse_rat.abs_eq
thf(fact_9829_Rat_Opositive_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( positive @ ( abs_Rat @ X ) )
        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).

% Rat.positive.abs_eq
thf(fact_9830_ratrel__iff,axiom,
    ( ratrel
    = ( ^ [X4: product_prod_int_int,Y3: product_prod_int_int] :
          ( ( ( product_snd_int_int @ X4 )
           != zero_zero_int )
          & ( ( product_snd_int_int @ Y3 )
           != zero_zero_int )
          & ( ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) )
            = ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ) ).

% ratrel_iff
thf(fact_9831_one__rat_Orsp,axiom,
    ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).

% one_rat.rsp
thf(fact_9832_zero__rat_Orsp,axiom,
    ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).

% zero_rat.rsp
thf(fact_9833_ratrel__def,axiom,
    ( ratrel
    = ( ^ [X4: product_prod_int_int,Y3: product_prod_int_int] :
          ( ( ( product_snd_int_int @ X4 )
           != zero_zero_int )
          & ( ( product_snd_int_int @ Y3 )
           != zero_zero_int )
          & ( ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) )
            = ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ) ).

% ratrel_def
thf(fact_9834_times__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
      ( ( ratrel @ Xa2 @ Xa2 )
     => ( ( ratrel @ X @ X )
       => ( ( times_times_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).

% times_rat.abs_eq
thf(fact_9835_min__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M2 @ N ) ) ) ).

% min_Suc_Suc
thf(fact_9836_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9837_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9838_min__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
      = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K2 ) ) ) ) ).

% min_Suc_numeral
thf(fact_9839_min__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K2 ) @ N ) ) ) ).

% min_numeral_Suc
thf(fact_9840_nat__mult__min__right,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ M2 @ ( ord_min_nat @ N @ Q3 ) )
      = ( ord_min_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q3 ) ) ) ).

% nat_mult_min_right
thf(fact_9841_nat__mult__min__left,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M2 @ N ) @ Q3 )
      = ( ord_min_nat @ ( times_times_nat @ M2 @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).

% nat_mult_min_left
thf(fact_9842_min__diff,axiom,
    ! [M2: nat,I: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M2 @ I ) @ ( minus_minus_nat @ N @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M2 @ N ) @ I ) ) ).

% min_diff
thf(fact_9843_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_9844_concat__bit__assoc__sym,axiom,
    ! [M2: nat,N: nat,K2: int,L: int,R2: int] :
      ( ( bit_concat_bit @ M2 @ ( bit_concat_bit @ N @ K2 @ L ) @ R2 )
      = ( bit_concat_bit @ ( ord_min_nat @ M2 @ N ) @ K2 @ ( bit_concat_bit @ ( minus_minus_nat @ M2 @ N ) @ L @ R2 ) ) ) ).

% concat_bit_assoc_sym
thf(fact_9845_take__bit__concat__bit__eq,axiom,
    ! [M2: nat,N: nat,K2: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ M2 @ ( bit_concat_bit @ N @ K2 @ L ) )
      = ( bit_concat_bit @ ( ord_min_nat @ M2 @ N ) @ K2 @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M2 @ N ) @ L ) ) ) ).

% take_bit_concat_bit_eq
thf(fact_9846_min__Suc1,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_min_nat @ ( suc @ N ) @ M2 )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ N @ M5 ) )
        @ M2 ) ) ).

% min_Suc1
thf(fact_9847_min__Suc2,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_min_nat @ M2 @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ M5 @ N ) )
        @ M2 ) ) ).

% min_Suc2
thf(fact_9848_min__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q3 )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(3)
thf(fact_9849_min__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ Q3 @ zero_z5237406670263579293d_enat )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(2)
thf(fact_9850_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
thf(fact_9851_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_9852_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M3: nat,N3: nat] :
          ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N3 ) @ M3 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9853_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( gcd_gcd_nat @ M2 @ N )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D2: nat] :
                ( ( dvd_dvd_nat @ D2 @ M2 )
                & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_9854_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_9855_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_9856_list__encode_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X5: nat,Xs3: list_nat] :
            ( X
           != ( cons_nat @ X5 @ Xs3 ) ) ) ).

% list_encode.cases
thf(fact_9857_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_9858_Inf__real__def,axiom,
    ( comple4887499456419720421f_real
    = ( ^ [X6: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X6 ) ) ) ) ) ).

% Inf_real_def
thf(fact_9859_list__encode__eq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( nat_list_encode @ X )
        = ( nat_list_encode @ Y ) )
      = ( X = Y ) ) ).

% list_encode_eq
thf(fact_9860_list__encode_Osimps_I1_J,axiom,
    ( ( nat_list_encode @ nil_nat )
    = zero_zero_nat ) ).

% list_encode.simps(1)
thf(fact_9861_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X @ Xs2 ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_9862_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K2: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K2 ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K2 ) ) @ ( cons_nat @ K2 @ nil_nat ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_9863_sorted__list__of__set__atMost__Suc,axiom,
    ! [K2: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K2 ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K2 ) ) @ ( cons_nat @ ( suc @ K2 ) @ nil_nat ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_9864_range__mod,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( image_nat_nat
          @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% range_mod
thf(fact_9865_UNIV__nat__eq,axiom,
    ( top_top_set_nat
    = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).

% UNIV_nat_eq
thf(fact_9866_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_9867_card__UNIV__bool,axiom,
    ( ( finite_card_o @ top_top_set_o )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% card_UNIV_bool
thf(fact_9868_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
      & ( ( A != zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_9869_surj__list__encode,axiom,
    ( ( image_list_nat_nat @ nat_list_encode @ top_top_set_list_nat )
    = top_top_set_nat ) ).

% surj_list_encode
thf(fact_9870_bij__list__encode,axiom,
    bij_be8532844293280997160at_nat @ nat_list_encode @ top_top_set_list_nat @ top_top_set_nat ).

% bij_list_encode
thf(fact_9871_surj__prod__encode,axiom,
    ( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
    = top_top_set_nat ) ).

% surj_prod_encode
thf(fact_9872_bij__prod__encode,axiom,
    bij_be5333170631980326235at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat @ top_top_set_nat ).

% bij_prod_encode
thf(fact_9873_root__def,axiom,
    ( root
    = ( ^ [N3: nat,X4: real] :
          ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y3: real] : ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N3 ) )
            @ X4 ) ) ) ) ).

% root_def
thf(fact_9874_card__UNIV__char,axiom,
    ( ( finite_card_char @ top_top_set_char )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% card_UNIV_char
thf(fact_9875_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_9876_char_Osize_I2_J,axiom,
    ! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_size_char @ ( char2 @ X1 @ X2 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_9877_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_9878_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_9879_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
      ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).

% integer_of_char_code
thf(fact_9880_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9881_String_Ochar__of__ascii__of,axiom,
    ! [C: char] :
      ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
      = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).

% String.char_of_ascii_of
thf(fact_9882_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_9883_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_9884_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_9885_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_9886_upto__single,axiom,
    ! [I: int] :
      ( ( upto @ I @ I )
      = ( cons_int @ I @ nil_int ) ) ).

% upto_single
thf(fact_9887_nth__upto,axiom,
    ! [I: int,K2: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K2 ) ) @ J )
     => ( ( nth_int @ ( upto @ I @ J ) @ K2 )
        = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K2 ) ) ) ) ).

% nth_upto
thf(fact_9888_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_9889_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_9890_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_9891_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_9892_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_9893_upto__code,axiom,
    ( upto
    = ( ^ [I3: int,J3: int] : ( upto_aux @ I3 @ J3 @ nil_int ) ) ) ).

% upto_code
thf(fact_9894_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I3: int,J3: int] : ( append_int @ ( upto @ I3 @ J3 ) ) ) ) ).

% upto_aux_def
thf(fact_9895_distinct__upto,axiom,
    ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).

% distinct_upto
thf(fact_9896_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ J3 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_9897_upto__split2,axiom,
    ! [I: int,J: int,K2: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K2 )
       => ( ( upto @ I @ K2 )
          = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K2 ) ) ) ) ) ).

% upto_split2
thf(fact_9898_upto__split1,axiom,
    ! [I: int,J: int,K2: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K2 )
       => ( ( upto @ I @ K2 )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K2 ) ) ) ) ) ).

% upto_split1
thf(fact_9899_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_9900_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_9901_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_9902_upto_Osimps,axiom,
    ( upto
    = ( ^ [I3: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J3 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_9903_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_9904_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_9905_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_9906_upto__split3,axiom,
    ! [I: int,J: int,K2: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K2 )
       => ( ( upto @ I @ K2 )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K2 ) ) ) ) ) ) ).

% upto_split3
thf(fact_9907_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_9908_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( X != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
             => ( ( ord_less_real @ X @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
               => ( D4
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9909_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( ord_less_real @ X @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9910_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9911_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9912_MVT2,axiom,
    ! [A: real,B: real,F: real > real,F4: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B )
             => ( has_fi5821293074295781190e_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
       => ? [Z3: real] :
            ( ( ord_less_real @ A @ Z3 )
            & ( ord_less_real @ Z3 @ B )
            & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F4 @ Z3 ) ) ) ) ) ) ).

% MVT2
thf(fact_9913_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9914_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9915_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: real > real,K2: real] :
      ( ( A != B )
     => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ K2 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
          = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K2 ) ) ) ) ).

% DERIV_const_ratio_const
thf(fact_9916_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: real > real,K2: real] :
      ( ( A != B )
     => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ K2 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
          = K2 ) ) ) ).

% DERIV_const_ratio_const2
thf(fact_9917_DERIV__local__const,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
             => ( ( F @ X )
                = ( F @ Y4 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9918_DERIV__const__average,axiom,
    ! [A: real,B: real,V: real > real,K2: real] :
      ( ( A != B )
     => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ V @ K2 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% DERIV_const_average
thf(fact_9919_DERIV__local__min,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9920_DERIV__local__max,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9921_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_ln_divide
thf(fact_9922_DERIV__pow,axiom,
    ! [N: nat,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X4: real] : ( power_power_real @ X4 @ N )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S2 ) ) ).

% DERIV_pow
thf(fact_9923_DERIV__fun__pow,axiom,
    ! [G: real > real,M2: real,X: real,N: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X4: real] : ( power_power_real @ ( G @ X4 ) @ N )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M2 )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9924_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z2: real] : ( powr_real @ Z2 @ R2 )
        @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_9925_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_9926_DERIV__fun__powr,axiom,
    ! [G: real > real,M2: real,X: real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X4: real] : ( powr_real @ ( G @ X4 ) @ R2 )
          @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M2 )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9927_DERIV__powr,axiom,
    ! [G: real > real,M2: real,X: real,F: real > real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X4: real] : ( powr_real @ ( G @ X4 ) @ ( F @ X4 ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M2 @ ( F @ X ) ) @ ( G @ X ) ) ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9928_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_real_sqrt
thf(fact_9929_DERIV__series_H,axiom,
    ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
      ( ! [N2: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X4: real] : ( F @ X4 @ N2 )
          @ ( F4 @ X0 @ N2 )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B ) )
           => ( summable_real @ ( F @ X5 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ( summable_real @ ( F4 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N2: nat,X5: real,Y4: real] :
                    ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B ) )
                   => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X5 @ N2 ) @ ( F @ Y4 @ N2 ) ) ) @ ( times_times_real @ ( L5 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X5 @ Y4 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X4: real] : ( suminf_real @ ( F @ X4 ) )
                  @ ( suminf_real @ ( F4 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_9930_DERIV__arctan,axiom,
    ! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).

% DERIV_arctan
thf(fact_9931_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A4: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A4 ) ) ).

% arsinh_real_has_field_derivative
thf(fact_9932_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X )
         => ( D4
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X @ zero_zero_real )
           => ( D4
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9933_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A4: set_real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A4 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_9934_artanh__real__has__field__derivative,axiom,
    ! [X: real,A4: set_real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A4 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9935_DERIV__power__series_H,axiom,
    ! [R: real,F: nat > real,X0: real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
         => ( summable_real
            @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X5 @ N3 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
       => ( ( ord_less_real @ zero_zero_real @ R )
         => ( has_fi5821293074295781190e_real
            @ ^ [X4: real] :
                ( suminf_real
                @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X4 @ ( suc @ N3 ) ) ) )
            @ ( suminf_real
              @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X0 @ N3 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9936_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9937_DERIV__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_arccos
thf(fact_9938_DERIV__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_arcsin
thf(fact_9939_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
     => ? [T5: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
          & ( ( F @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9940_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ? [T5: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9941_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( X != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9942_Maclaurin,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M: nat,T5: real] :
                ( ( ( ord_less_nat @ M @ N )
                  & ( ord_less_eq_real @ zero_zero_real @ T5 )
                  & ( ord_less_eq_real @ T5 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
           => ? [T5: real] :
                ( ( ord_less_real @ zero_zero_real @ T5 )
                & ( ord_less_real @ T5 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9943_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M: nat,T5: real] :
              ( ( ( ord_less_nat @ M @ N )
                & ( ord_less_eq_real @ zero_zero_real @ T5 )
                & ( ord_less_eq_real @ T5 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
         => ? [T5: real] :
              ( ( ord_less_real @ zero_zero_real @ T5 )
              & ( ord_less_eq_real @ T5 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
                    @ ( set_ord_lessThan_nat @ N ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9944_Maclaurin__minus,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M: nat,T5: real] :
                ( ( ( ord_less_nat @ M @ N )
                  & ( ord_less_eq_real @ H2 @ T5 )
                  & ( ord_less_eq_real @ T5 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
           => ? [T5: real] :
                ( ( ord_less_real @ H2 @ T5 )
                & ( ord_less_real @ T5 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9945_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( X != zero_zero_real )
         => ( ! [M: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
           => ? [T5: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
                & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
                & ( ( F @ X )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9946_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M: nat,T5: real] :
            ( ( ( ord_less_nat @ M @ N )
              & ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
       => ? [T5: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9947_Taylor,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M: nat,T5: real] :
              ( ( ( ord_less_nat @ M @ N )
                & ( ord_less_eq_real @ A @ T5 )
                & ( ord_less_eq_real @ T5 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ( ( ord_less_eq_real @ A @ X )
               => ( ( ord_less_eq_real @ X @ B )
                 => ( ( X != C )
                   => ? [T5: real] :
                        ( ( ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ X @ T5 )
                            & ( ord_less_real @ T5 @ C ) ) )
                        & ( ~ ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ C @ T5 )
                            & ( ord_less_real @ T5 @ X ) ) )
                        & ( ( F @ X )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M3 ) )
                              @ ( set_ord_lessThan_nat @ N ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9948_Taylor__up,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M: nat,T5: real] :
              ( ( ( ord_less_nat @ M @ N )
                & ( ord_less_eq_real @ A @ T5 )
                & ( ord_less_eq_real @ T5 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B )
             => ? [T5: real] :
                  ( ( ord_less_real @ C @ T5 )
                  & ( ord_less_real @ T5 @ B )
                  & ( ( F @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M3 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9949_Taylor__down,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M: nat,T5: real] :
              ( ( ( ord_less_nat @ M @ N )
                & ( ord_less_eq_real @ A @ T5 )
                & ( ord_less_eq_real @ T5 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ? [T5: real] :
                  ( ( ord_less_real @ A @ T5 )
                  & ( ord_less_real @ T5 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M3 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9950_Maclaurin__lemma2,axiom,
    ! [N: nat,H2: real,Diff: nat > real > real,K2: nat,B4: real] :
      ( ! [M: nat,T5: real] :
          ( ( ( ord_less_nat @ M @ N )
            & ( ord_less_eq_real @ zero_zero_real @ T5 )
            & ( ord_less_eq_real @ T5 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
     => ( ( N
          = ( suc @ K2 ) )
       => ! [M4: nat,T6: real] :
            ( ( ( ord_less_nat @ M4 @ N )
              & ( ord_less_eq_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M4 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M4 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M4 ) ) )
                    @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M4 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M4 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M4 ) @ T6 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M4 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T6 @ P5 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M4 ) ) ) )
                  @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ T6 @ ( minus_minus_nat @ N @ ( suc @ M4 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M4 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9951_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X10: real] :
            ( suminf_real
            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_9952_LIM__fun__less__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X3 ) ) @ R4 ) )
               => ( ord_less_real @ ( F @ X3 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_9953_LIM__fun__not__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( L != zero_zero_real )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X3 ) ) @ R4 ) )
               => ( ( F @ X3 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_9954_LIM__fun__gt__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X3 ) ) @ R4 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X3 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_9955_isCont__arcosh,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).

% isCont_arcosh
thf(fact_9956_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X4: real] : ( divide_divide_real @ ( cos_real @ X4 ) @ ( sin_real @ X4 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).

% LIM_cos_div_sin
thf(fact_9957_isCont__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).

% isCont_arccos
thf(fact_9958_isCont__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).

% isCont_arcsin
thf(fact_9959_isCont__artanh,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).

% isCont_artanh
thf(fact_9960_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
           => ( ( G @ ( F @ Z3 ) )
              = Z3 ) )
       => ( ! [Z3: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_9961_GMVT_H,axiom,
    ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [Z3: real] :
            ( ( ord_less_eq_real @ A @ Z3 )
           => ( ( ord_less_eq_real @ Z3 @ B )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z3: real] :
              ( ( ord_less_eq_real @ A @ Z3 )
             => ( ( ord_less_eq_real @ Z3 @ B )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z3: real] :
                ( ( ord_less_real @ A @ Z3 )
               => ( ( ord_less_real @ Z3 @ B )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
           => ( ! [Z3: real] :
                  ( ( ord_less_real @ A @ Z3 )
                 => ( ( ord_less_real @ Z3 @ B )
                   => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
             => ? [C2: real] :
                  ( ( ord_less_real @ A @ C2 )
                  & ( ord_less_real @ C2 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C2 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F4 @ C2 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_9962_summable__Leibniz_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
         => ! [N9: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_9963_summable__Leibniz_I2_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
         => ! [N9: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_9964_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_9965_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X4: nat] : ( times_times_nat @ X4 @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_9966_LIMSEQ__root,axiom,
    ( filterlim_nat_real
    @ ^ [N3: nat] : ( root @ N3 @ ( semiri5074537144036343181t_real @ N3 ) )
    @ ( topolo2815343760600316023s_real @ one_one_real )
    @ at_top_nat ) ).

% LIMSEQ_root
thf(fact_9967_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
         => ( ( filterlim_nat_real
              @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L4: real] :
                ( ! [N9: nat] : ( ord_less_eq_real @ ( F @ N9 ) @ L4 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
                & ! [N9: nat] : ( ord_less_eq_real @ L4 @ ( G @ N9 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_9968_LIMSEQ__inverse__zero,axiom,
    ! [X9: nat > real] :
      ( ! [R4: real] :
        ? [N7: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N7 @ N2 )
         => ( ord_less_real @ R4 @ ( X9 @ N2 ) ) )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( inverse_inverse_real @ ( X9 @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_9969_lim__inverse__n_H,axiom,
    ( filterlim_nat_real
    @ ^ [N3: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% lim_inverse_n'
thf(fact_9970_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( root @ N3 @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_9971_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N3: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_9972_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_9973_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L: real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
       => ( ! [E2: real] :
              ( ( ord_less_real @ zero_zero_real @ E2 )
             => ? [N9: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N9 ) @ E2 ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_9974_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_9975_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_9976_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero2
thf(fact_9977_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero
thf(fact_9978_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_9979_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9980_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ N3 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ at_top_nat ) ).

% tendsto_exp_limit_sequentially
thf(fact_9981_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9982_summable__Leibniz_I1_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_9983_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( summable_real
            @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ) ).

% summable
thf(fact_9984_cos__diff__limit__1,axiom,
    ! [Theta: nat > real,Theta2: real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ~ ! [K: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_9985_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K: nat > int] :
          ( filterlim_nat_real
          @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_9986_summable__Leibniz_I4_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_9987_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_9988_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_9989_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_9990_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L4: real] :
              ( ! [N9: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N3: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N9: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N3: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_9991_summable__Leibniz_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_9992_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_9993_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_9994_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_9995_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K2: nat] :
      ( ( eventually_nat
        @ ^ [N3: nat] : ( P @ ( plus_plus_nat @ N3 @ K2 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_9996_le__sequentially,axiom,
    ! [F5: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
      = ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F5 ) ) ) ).

% le_sequentially
thf(fact_9997_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X5: nat] :
          ( ( ord_less_eq_nat @ C @ X5 )
         => ( P @ X5 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_9998_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N6: nat] :
          ! [N3: nat] :
            ( ( ord_less_eq_nat @ N6 @ N3 )
           => ( P @ N3 ) ) ) ) ).

% eventually_sequentially
thf(fact_9999_sequentially__offset,axiom,
    ! [P: nat > $o,K2: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I3: nat] : ( P @ ( plus_plus_nat @ I3 @ K2 ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_10000_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_10001_real__bounded__linear,axiom,
    ( real_V5970128139526366754l_real
    = ( ^ [F3: real > real] :
        ? [C3: real] :
          ( F3
          = ( ^ [X4: real] : ( times_times_real @ X4 @ C3 ) ) ) ) ) ).

% real_bounded_linear
thf(fact_10002_tanh__real__at__top,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).

% tanh_real_at_top
thf(fact_10003_artanh__real__at__left__1,axiom,
    filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).

% artanh_real_at_left_1
thf(fact_10004_tendsto__exp__limit__at__top,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y3: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y3 ) ) @ Y3 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ at_top_real ) ).

% tendsto_exp_limit_at_top
thf(fact_10005_filterlim__tan__at__left,axiom,
    filterlim_real_real @ tan_real @ at_top_real @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( set_or5984915006950818249n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% filterlim_tan_at_left
thf(fact_10006_tendsto__arctan__at__top,axiom,
    filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).

% tendsto_arctan_at_top
thf(fact_10007_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: real > real,F5: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N )
            @ at_top_real
            @ F5 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_10008_dist__complex__def,axiom,
    ( real_V3694042436643373181omplex
    = ( ^ [X4: complex,Y3: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y3 ) ) ) ) ).

% dist_complex_def
thf(fact_10009_dist__real__def,axiom,
    ( real_V975177566351809787t_real
    = ( ^ [X4: real,Y3: real] : ( abs_abs_real @ ( minus_minus_real @ X4 @ Y3 ) ) ) ) ).

% dist_real_def
thf(fact_10010_tanh__real__at__bot,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).

% tanh_real_at_bot
thf(fact_10011_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: real > real,F5: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N )
            @ at_bot_real
            @ F5 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_10012_tendsto__arctan__at__bot,axiom,
    filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).

% tendsto_arctan_at_bot
thf(fact_10013_tendsto__exp__limit__at__right,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y3: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y3 ) ) @ ( divide_divide_real @ one_one_real @ Y3 ) )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% tendsto_exp_limit_at_right
thf(fact_10014_filterlim__tan__at__right,axiom,
    filterlim_real_real @ tan_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% filterlim_tan_at_right
thf(fact_10015_tendsto__arcosh__at__left__1,axiom,
    filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).

% tendsto_arcosh_at_left_1
thf(fact_10016_artanh__real__at__right__1,axiom,
    filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% artanh_real_at_right_1
thf(fact_10017_atLeast__Suc__greaterThan,axiom,
    ! [K2: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K2 ) )
      = ( set_or1210151606488870762an_nat @ K2 ) ) ).

% atLeast_Suc_greaterThan
thf(fact_10018_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_10019_greaterThan__Suc,axiom,
    ! [K2: nat] :
      ( ( set_or1210151606488870762an_nat @ ( suc @ K2 ) )
      = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K2 ) @ ( insert_nat @ ( suc @ K2 ) @ bot_bot_set_nat ) ) ) ).

% greaterThan_Suc
thf(fact_10020_atLeast__Suc,axiom,
    ! [K2: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K2 ) )
      = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K2 ) @ ( insert_nat @ K2 @ bot_bot_set_nat ) ) ) ).

% atLeast_Suc
thf(fact_10021_GMVT,axiom,
    ! [A: real,B: real,F: real > real,G: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X5: real] :
            ( ( ( ord_less_eq_real @ A @ X5 )
              & ( ord_less_eq_real @ X5 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
       => ( ! [X5: real] :
              ( ( ( ord_less_real @ A @ X5 )
                & ( ord_less_real @ X5 @ B ) )
             => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
         => ( ! [X5: real] :
                ( ( ( ord_less_eq_real @ A @ X5 )
                  & ( ord_less_eq_real @ X5 @ B ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ G ) )
           => ( ! [X5: real] :
                  ( ( ( ord_less_real @ A @ X5 )
                    & ( ord_less_real @ X5 @ B ) )
                 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
                  & ( ord_less_real @ A @ C2 )
                  & ( ord_less_real @ C2 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).

% GMVT
thf(fact_10022_MVT,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A @ X5 )
             => ( ( ord_less_real @ X5 @ B )
               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ? [L4: real,Z3: real] :
              ( ( ord_less_real @ A @ Z3 )
              & ( ord_less_real @ Z3 @ B )
              & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).

% MVT
thf(fact_10023_continuous__on__arcosh,axiom,
    ! [A4: set_real] :
      ( ( ord_less_eq_set_real @ A4 @ ( set_ord_atLeast_real @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A4 @ arcosh_real ) ) ).

% continuous_on_arcosh
thf(fact_10024_continuous__on__arcosh_H,axiom,
    ! [A4: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A4 @ F )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A4 )
           => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
       => ( topolo5044208981011980120l_real @ A4
          @ ^ [X4: real] : ( arcosh_real @ ( F @ X4 ) ) ) ) ) ).

% continuous_on_arcosh'
thf(fact_10025_continuous__on__arccos_H,axiom,
    topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).

% continuous_on_arccos'
thf(fact_10026_continuous__on__arcsin_H,axiom,
    topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).

% continuous_on_arcsin'
thf(fact_10027_continuous__on__artanh,axiom,
    ! [A4: set_real] :
      ( ( ord_less_eq_set_real @ A4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A4 @ artanh_real ) ) ).

% continuous_on_artanh
thf(fact_10028_continuous__on__artanh_H,axiom,
    ! [A4: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A4 @ F )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A4 )
           => ( member_real @ ( F @ X5 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
       => ( topolo5044208981011980120l_real @ A4
          @ ^ [X4: real] : ( artanh_real @ ( F @ X4 ) ) ) ) ) ).

% continuous_on_artanh'
thf(fact_10029_mvt,axiom,
    ! [A: real,B: real,F: real > real,F4: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A @ X5 )
             => ( ( ord_less_real @ X5 @ B )
               => ( has_de1759254742604945161l_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ~ ! [Xi: real] :
                ( ( ord_less_real @ A @ Xi )
               => ( ( ord_less_real @ Xi @ B )
                 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                   != ( F4 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).

% mvt
thf(fact_10030_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_10031_mono__times__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).

% mono_times_nat
thf(fact_10032_Bseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).

% Bseq_realpow
thf(fact_10033_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
     => ( order_mono_nat_nat
        @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K2 @ M3 ) @ M3 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_10034_inj__sgn__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( inj_on_real_real
        @ ^ [Y3: real] : ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_10035_log__inj,axiom,
    ! [B: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% log_inj
thf(fact_10036_inj__list__encode,axiom,
    ! [A4: set_list_nat] : ( inj_on_list_nat_nat @ nat_list_encode @ A4 ) ).

% inj_list_encode
thf(fact_10037_inj__Suc,axiom,
    ! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).

% inj_Suc
thf(fact_10038_inj__prod__encode,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : ( inj_on2178005380612969504at_nat @ nat_prod_encode @ A4 ) ).

% inj_prod_encode
thf(fact_10039_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K2: nat] :
      ( ! [N2: nat] :
          ( ( member_nat @ N2 @ N5 )
         => ( ord_less_eq_nat @ K2 @ N2 ) )
     => ( inj_on_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K2 )
        @ N5 ) ) ).

% inj_on_diff_nat
thf(fact_10040_inj__on__set__encode,axiom,
    inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).

% inj_on_set_encode
thf(fact_10041_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_10042_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_10043_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
thf(fact_10044_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K2 ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K2 ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_10045_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_10046_remdups__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( remdups_nat @ ( upt @ M2 @ N ) )
      = ( upt @ M2 @ N ) ) ).

% remdups_upt
thf(fact_10047_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_10048_drop__upt,axiom,
    ! [M2: nat,I: nat,J: nat] :
      ( ( drop_nat @ M2 @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M2 ) @ J ) ) ).

% drop_upt
thf(fact_10049_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_10050_take__upt,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M2 ) @ N )
     => ( ( take_nat @ M2 @ ( upt @ I @ N ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M2 ) ) ) ) ).

% take_upt
thf(fact_10051_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_10052_sorted__list__of__set__range,axiom,
    ! [M2: nat,N: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
      = ( upt @ M2 @ N ) ) ).

% sorted_list_of_set_range
thf(fact_10053_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_10054_nth__upt,axiom,
    ! [I: nat,K2: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J )
     => ( ( nth_nat @ ( upt @ I @ J ) @ K2 )
        = ( plus_plus_nat @ I @ K2 ) ) ) ).

% nth_upt
thf(fact_10055_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_10056_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).

% atMost_upto
thf(fact_10057_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N3 ) ) ) ) ).

% atLeast_upt
thf(fact_10058_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_10059_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q3: nat] :
      ( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M2 @ Q3 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M2 ) @ Q3 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_10060_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ M3 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_10061_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ ( suc @ M3 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_10062_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I3: nat,J3: nat] : ( set_nat2 @ ( upt @ I3 @ J3 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_10063_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ N3 @ ( suc @ M3 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_10064_distinct__upt,axiom,
    ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).

% distinct_upt
thf(fact_10065_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_10066_map__add__upt,axiom,
    ! [N: nat,M2: nat] :
      ( ( map_nat_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N )
        @ ( upt @ zero_zero_nat @ M2 ) )
      = ( upt @ N @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% map_add_upt
thf(fact_10067_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_10068_map__decr__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( upt @ M2 @ N ) ) ).

% map_decr_upt
thf(fact_10069_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus_nat @ J @ K2 ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K2 ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_10070_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs2: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X @ Xs2 ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs2 ) ) ) ).

% upt_eq_Cons_conv
thf(fact_10071_upt__rec,axiom,
    ( upt
    = ( ^ [I3: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J3 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_10072_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_10073_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_10074_tl__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ N ) ) ).

% tl_upt
thf(fact_10075_sum__list__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M2 @ N ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ) ).

% sum_list_upt
thf(fact_10076_card__length__sum__list__rec,axiom,
    ! [M2: nat,N5: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M2 )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L2: list_nat] :
                ( ( ( size_size_list_nat @ L2 )
                  = M2 )
                & ( ( groups4561878855575611511st_nat @ L2 )
                  = N5 ) ) ) )
        = ( plus_plus_nat
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = ( minus_minus_nat @ M2 @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L2 )
                    = N5 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = M2 )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
                    = N5 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_10077_card__length__sum__list,axiom,
    ! [M2: nat,N5: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L2: list_nat] :
              ( ( ( size_size_list_nat @ L2 )
                = M2 )
              & ( ( groups4561878855575611511st_nat @ L2 )
                = N5 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M2 ) @ one_one_nat ) @ N5 ) ) ).

% card_length_sum_list
thf(fact_10078_sorted__upto,axiom,
    ! [M2: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M2 @ N ) ) ).

% sorted_upto
thf(fact_10079_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).

% sorted_wrt_upto
thf(fact_10080_sorted__wrt__upt,axiom,
    ! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M2 @ N ) ) ).

% sorted_wrt_upt
thf(fact_10081_sorted__upt,axiom,
    ! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M2 @ N ) ) ).

% sorted_upt
thf(fact_10082_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_10083_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M3: nat,N3: nat] :
            ( N3
            = ( suc @ M3 ) ) ) ) ) ).

% pred_nat_def
thf(fact_10084_less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% less_eq
thf(fact_10085_pred__nat__trancl__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% pred_nat_trancl_eq_le
thf(fact_10086_product__atMost__eq__Un,axiom,
    ! [A4: set_nat,M2: nat] :
      ( ( produc457027306803732586at_nat @ A4
        @ ^ [Uu3: nat] : ( set_ord_atMost_nat @ M2 ) )
      = ( sup_su6327502436637775413at_nat
        @ ( produc457027306803732586at_nat @ A4
          @ ^ [I3: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M2 @ I3 ) ) )
        @ ( produc457027306803732586at_nat @ A4
          @ ^ [I3: nat] : ( set_or6659071591806873216st_nat @ ( minus_minus_nat @ M2 @ I3 ) @ M2 ) ) ) ) ).

% product_atMost_eq_Un
thf(fact_10087_pairs__le__eq__Sigma,axiom,
    ! [M2: nat] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J3 ) @ M2 ) ) )
      = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M2 )
        @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M2 @ R5 ) ) ) ) ).

% pairs_le_eq_Sigma
thf(fact_10088_coprime__Suc__right__nat,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ N ) ) ).

% coprime_Suc_right_nat
thf(fact_10089_coprime__Suc__left__nat,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ N ) @ N ) ).

% coprime_Suc_left_nat
thf(fact_10090_coprime__crossproduct__nat,axiom,
    ! [A: nat,D: nat,B: nat,C: nat] :
      ( ( algebr934650988132801477me_nat @ A @ D )
     => ( ( algebr934650988132801477me_nat @ B @ C )
       => ( ( ( times_times_nat @ A @ C )
            = ( times_times_nat @ B @ D ) )
          = ( ( A = B )
            & ( C = D ) ) ) ) ) ).

% coprime_crossproduct_nat
thf(fact_10091_coprime__Suc__0__right,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).

% coprime_Suc_0_right
thf(fact_10092_coprime__Suc__0__left,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).

% coprime_Suc_0_left
thf(fact_10093_coprime__common__divisor__nat,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( ( algebr934650988132801477me_nat @ A @ B )
     => ( ( dvd_dvd_nat @ X @ A )
       => ( ( dvd_dvd_nat @ X @ B )
         => ( X = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
thf(fact_10094_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).

% coprime_diff_one_left_nat
thf(fact_10095_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% coprime_diff_one_right_nat
thf(fact_10096_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( ( member_real @ X @ field_5140801741446780682s_real )
     => ~ ! [M: nat,N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( ( ( abs_abs_real @ X )
                = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
             => ~ ( algebr934650988132801477me_nat @ M @ N2 ) ) ) ) ).

% Rats_abs_nat_div_natE
thf(fact_10097_coprime__abs__left__iff,axiom,
    ! [K2: int,L: int] :
      ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K2 ) @ L )
      = ( algebr932160517623751201me_int @ K2 @ L ) ) ).

% coprime_abs_left_iff
thf(fact_10098_coprime__abs__right__iff,axiom,
    ! [K2: int,L: int] :
      ( ( algebr932160517623751201me_int @ K2 @ ( abs_abs_int @ L ) )
      = ( algebr932160517623751201me_int @ K2 @ L ) ) ).

% coprime_abs_right_iff
thf(fact_10099_coprime__int__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( algebr934650988132801477me_nat @ M2 @ N ) ) ).

% coprime_int_iff
thf(fact_10100_coprime__nat__abs__left__iff,axiom,
    ! [K2: int,N: nat] :
      ( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ N )
      = ( algebr932160517623751201me_int @ K2 @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% coprime_nat_abs_left_iff
thf(fact_10101_coprime__nat__abs__right__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K2 ) ) )
      = ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K2 ) ) ).

% coprime_nat_abs_right_iff
thf(fact_10102_coprime__common__divisor__int,axiom,
    ! [A: int,B: int,X: int] :
      ( ( algebr932160517623751201me_int @ A @ B )
     => ( ( dvd_dvd_int @ X @ A )
       => ( ( dvd_dvd_int @ X @ B )
         => ( ( abs_abs_int @ X )
            = one_one_int ) ) ) ) ).

% coprime_common_divisor_int
thf(fact_10103_coprime__crossproduct__int,axiom,
    ! [A: int,D: int,B: int,C: int] :
      ( ( algebr932160517623751201me_int @ A @ D )
     => ( ( algebr932160517623751201me_int @ B @ C )
       => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
            = ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D ) ) )
          = ( ( ( abs_abs_int @ A )
              = ( abs_abs_int @ B ) )
            & ( ( abs_abs_int @ C )
              = ( abs_abs_int @ D ) ) ) ) ) ) ).

% coprime_crossproduct_int
thf(fact_10104_vimage__Suc__insert__Suc,axiom,
    ! [N: nat,A4: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A4 ) )
      = ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A4 ) ) ) ).

% vimage_Suc_insert_Suc
thf(fact_10105_vimage__Suc__insert__0,axiom,
    ! [A4: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A4 ) )
      = ( vimage_nat_nat @ suc @ A4 ) ) ).

% vimage_Suc_insert_0
thf(fact_10106_finite__vimage__Suc__iff,axiom,
    ! [F5: set_nat] :
      ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F5 ) )
      = ( finite_finite_nat @ F5 ) ) ).

% finite_vimage_Suc_iff
thf(fact_10107_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_10108_set__encode__vimage__Suc,axiom,
    ! [A4: set_nat] :
      ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A4 ) )
      = ( divide_divide_nat @ ( nat_set_encode @ A4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% set_encode_vimage_Suc
thf(fact_10109_Field__natLeq__on,axiom,
    ! [N: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y3: nat] :
                ( ( ord_less_nat @ X4 @ N )
                & ( ord_less_nat @ Y3 @ N )
                & ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) )
      = ( collect_nat
        @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) ) ) ).

% Field_natLeq_on
thf(fact_10110_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less

% Helper facts (39)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__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_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
      = ( some_nat @ sxa ) )
    = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) ) @ xa @ sxa ) ) ).

%------------------------------------------------------------------------------