%------------------------------------------------------------------------------
% File     : ITP284^1 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_BuildupMemImp 00837_038274
% 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    : 0093_VEBT_BuildupMemImp_00837_038274 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11356 (5777 unt;1109 typ;   0 def)
%            Number of atoms       : 29029 (13511 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 131996 (2866   ~; 499   |;1817   &;115706   @)
%                                         (   0 <=>;11108  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :  129 ( 128 usr)
%            Number of type conns  : 4290 (4290   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  984 ( 981 usr;  66 con; 0-8 aty)
%            Number of variables   : 26930 (1492   ^;24566   !; 872   ?;26930   :)
% 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 19:14:48.409
%------------------------------------------------------------------------------
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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 (981)
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_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_Array__Time_Ofreeze_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_8141364883501958055_VEBTi: array_VEBT_VEBTi > heap_T4980287057938770641_VEBTi ).

thf(sy_c_Array__Time_Onth_001_Eo,type,
    array_nth_o: array_o > nat > heap_Time_Heap_o ).

thf(sy_c_Array__Time_Onth_001t__Int__Oint,type,
    array_nth_int: array_int > nat > heap_Time_Heap_int ).

thf(sy_c_Array__Time_Onth_001t__Nat__Onat,type,
    array_nth_nat: array_nat > nat > heap_Time_Heap_nat ).

thf(sy_c_Array__Time_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
    array_nth_option_nat: array_option_nat > nat > heap_T2636463487746394924on_nat ).

thf(sy_c_Array__Time_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_nth_VEBT_VEBTi: array_VEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: assn > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Oentails,type,
    entails: assn > assn > $o ).

thf(sy_c_Assertions_Oex__assn_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    ex_ass463751140784270563_VEBTi: ( list_VEBT_VEBTi > assn ) > assn ).

thf(sy_c_Assertions_Opure__assn,type,
    pure_assn: $o > assn ).

thf(sy_c_Assertions_Osnga__assn_001_Eo,type,
    snga_assn_o: array_o > list_o > assn ).

thf(sy_c_Assertions_Osnga__assn_001t__Int__Oint,type,
    snga_assn_int: array_int > list_int > assn ).

thf(sy_c_Assertions_Osnga__assn_001t__Nat__Onat,type,
    snga_assn_nat: array_nat > list_nat > assn ).

thf(sy_c_Assertions_Osnga__assn_001t__Option__Ooption_It__Nat__Onat_J,type,
    snga_assn_option_nat: array_option_nat > list_option_nat > assn ).

thf(sy_c_Assertions_Osnga__assn_001t__VEBT____BuildupMemImp__OVEBTi,type,
    snga_assn_VEBT_VEBTi: array_VEBT_VEBTi > list_VEBT_VEBTi > assn ).

thf(sy_c_Automation_OFI__QUERY,type,
    fI_QUERY: assn > assn > assn > $o ).

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

thf(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Int__Oint,type,
    bit_bi6516823479961619367ts_int: ( nat > $o ) > int ).

thf(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
    bit_wf_set_bits_int: ( nat > $o ) > $o ).

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__Code____Numeral__Ointeger,type,
    bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).

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

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

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Ointeger,type,
    bit_se3928097537394005634nteger: nat > code_integer > code_integer ).

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__Code____Numeral__Ointeger,type,
    bit_se7788150548672797655nteger: nat > code_integer > code_integer ).

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Ointeger,type,
    bit_se9216721137139052372nteger: code_integer > nat > $o ).

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

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

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

thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl_001t__Nat__Onat,type,
    bit_Sh3965577149348748681tl_nat: nat > nat > nat ).

thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr_001t__Nat__Onat,type,
    bit_Sh2154871086232339855tr_nat: nat > nat > nat ).

thf(sy_c_Bits__Integer_OBit__integer,type,
    bits_Bit_integer: code_integer > $o > code_integer ).

thf(sy_c_Bits__Integer_Obin__last__integer,type,
    bits_b8758750999018896077nteger: code_integer > $o ).

thf(sy_c_Bits__Integer_Obin__rest__integer,type,
    bits_b2549910563261871055nteger: code_integer > code_integer ).

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

thf(sy_c_Code__Numeral_Odup,type,
    code_dup: code_integer > code_integer ).

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

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

thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
    has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
    semiri3624122377584611663nteger: nat > code_integer ).

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

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

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

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
    semiri773545260158071498ct_rat: nat > rat ).

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

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
    invers8013647133539491842omplex: complex > complex ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
    inverse_inverse_rat: rat > rat ).

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

thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
    at_bot_real: filter_real ).

thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
    at_top_nat: filter_nat ).

thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
    at_top_real: filter_real ).

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

thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
    eventually_real: ( real > $o ) > filter_real > $o ).

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

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

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

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

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

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

thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
    finite_card_nat: set_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Numeral____Type__Onum0,type,
    finite6454714172617411596l_num0: set_Numeral_num0 > nat ).

thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
    finite_card_char: set_char > nat ).

thf(sy_c_Finite__Set_Ocard_001t__String__Oliteral,type,
    finite_card_literal: set_literal > nat ).

thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
    finite_finite_o: set_o > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Code____Numeral__Ointeger,type,
    finite6017078050557962740nteger: set_Code_integer > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
    finite3207457112153483333omplex: set_complex > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
    finite_finite_int: set_int > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
    finite_finite_list_o: set_list_o > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
    finite8712137658972009173omplex: set_list_complex > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
    finite3922522038869484883st_int: set_list_int > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
    finite8100373058378681591st_nat: set_list_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Real__Oreal_J,type,
    finite306553202115118035t_real: set_list_real > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
    finite_finite_num: set_num > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    finite2998713641127702882nt_int: set_Pr958786334691620121nt_int > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
    finite_finite_rat: set_rat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
    finite_finite_real: set_real > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__VEBT____Definitions__OVEBT,type,
    finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).

thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).

thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).

thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
    bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001_Eo_001t__Nat__Onat,type,
    comp_nat_o_nat: ( nat > $o ) > ( nat > nat ) > nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
    inj_on_nat_char: ( nat > char ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
    inj_on_real_real: ( real > real ) > set_real > $o ).

thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
    the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
    abs_abs_Code_integer: code_integer > code_integer ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
    abs_abs_complex: complex > complex ).

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thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
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thf(sy_c_Heap_Oarray_Osize__array_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Heap__Time__Monad_Oreturn_001_Eo,type,
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thf(sy_c_Heap__Time__Monad_Oreturn_001t__Nat__Onat,type,
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thf(sy_c_Heap__Time__Monad_Oreturn_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001_Eo,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Nat__Onat,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_If_001t__Assertions__Oassn,type,
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thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_If_001t__Complex__Ocomplex,type,
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thf(sy_c_If_001t__Int__Oint,type,
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thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Num__Onum,type,
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thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__Rat__Orat,type,
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thf(sy_c_If_001t__Real__Oreal,type,
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thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Int_Oint__ge__less__than,type,
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thf(sy_c_Int_Oint__ge__less__than2,type,
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thf(sy_c_Int_Onat,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
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thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
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thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
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thf(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Int__Oint,type,
    least_4859182151741483524sb_int: int > $o ).

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_Ofilter_001t__Nat__Onat,type,
    filter_nat2: ( nat > $o ) > list_nat > list_nat ).

thf(sy_c_List_Ofoldr_001_Eo_001t__Nat__Onat,type,
    foldr_o_nat: ( $o > nat > nat ) > list_o > nat > nat ).

thf(sy_c_List_Ofoldr_001t__Int__Oint_001t__Nat__Onat,type,
    foldr_int_nat: ( int > nat > nat ) > list_int > nat > nat ).

thf(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
    foldr_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).

thf(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Nat__Onat,type,
    foldr_real_nat: ( real > nat > nat ) > list_real > nat > nat ).

thf(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Real__Oreal,type,
    foldr_real_real: ( real > real > real ) > list_real > real > real ).

thf(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
    linord1735203802627413978nt_int: ( int > int ) > list_int > list_int ).

thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
    linord738340561235409698at_nat: ( 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_001_Eo,type,
    cons_o: $o > list_o > list_o ).

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

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

thf(sy_c_List_Olist_ONil_001_Eo,type,
    nil_o: list_o ).

thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

thf(sy_c_List_Olist_Omap_001_Eo_001_Eo,type,
    map_o_o: ( $o > $o ) > list_o > list_o ).

thf(sy_c_List_Olist_Omap_001_Eo_001t__Nat__Onat,type,
    map_o_nat: ( $o > nat ) > list_o > list_nat ).

thf(sy_c_List_Olist_Omap_001_Eo_001t__Real__Oreal,type,
    map_o_real: ( $o > real ) > list_o > list_real ).

thf(sy_c_List_Olist_Omap_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    map_o_VEBT_VEBT: ( $o > vEBT_VEBT ) > list_o > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    map_complex_complex: ( complex > complex ) > list_complex > list_complex ).

thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    map_complex_real: ( complex > real ) > list_complex > list_real ).

thf(sy_c_List_Olist_Omap_001t__Int__Oint_001_Eo,type,
    map_int_o: ( int > $o ) > list_int > list_o ).

thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Int__Oint,type,
    map_int_int: ( int > int ) > list_int > list_int ).

thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Nat__Onat,type,
    map_int_nat: ( int > nat ) > list_int > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Real__Oreal,type,
    map_int_real: ( int > real ) > list_int > list_real ).

thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    map_int_VEBT_VEBT: ( int > vEBT_VEBT ) > list_int > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001_Eo,type,
    map_nat_o: ( nat > $o ) > list_nat > list_o ).

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_Omap_001t__Nat__Onat_001t__Real__Oreal,type,
    map_nat_real: ( nat > real ) > list_nat > list_real ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    map_nat_VEBT_VEBT: ( nat > vEBT_VEBT ) > list_nat > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001_Eo,type,
    map_real_o: ( real > $o ) > list_real > list_o ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
    map_real_nat: ( real > nat ) > list_real > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
    map_real_real: ( real > real ) > list_real > list_real ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    map_real_VEBT_VEBT: ( real > vEBT_VEBT ) > list_real > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    map_VEBT_VEBTi_int: ( vEBT_VEBTi > int ) > list_VEBT_VEBTi > list_int ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    map_VEBT_VEBTi_nat: ( vEBT_VEBTi > nat ) > list_VEBT_VEBTi > list_nat ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_VE483055756984248624_VEBTi: ( vEBT_VEBTi > vEBT_VEBTi ) > list_VEBT_VEBTi > list_VEBT_VEBTi ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    map_VE7998069337340375161T_VEBT: ( vEBT_VEBTi > vEBT_VEBT ) > list_VEBT_VEBTi > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    map_VEBT_VEBT_int: ( vEBT_VEBT > int ) > list_VEBT_VEBT > list_int ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    map_VEBT_VEBT_nat: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > list_nat ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    map_VEBT_VEBT_real: ( vEBT_VEBT > real ) > list_VEBT_VEBT > list_real ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_VE7029150624388687525_VEBTi: ( vEBT_VEBT > vEBT_VEBTi ) > list_VEBT_VEBT > list_VEBT_VEBTi ).

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

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

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

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

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

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

thf(sy_c_List_Olist_Oset_001t__VEBT____BuildupMemImp__OVEBTi,type,
    set_VEBT_VEBTi2: list_VEBT_VEBTi > set_VEBT_VEBTi ).

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

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

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

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

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

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

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

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

thf(sy_c_List_Olist__update_001t__VEBT____BuildupMemImp__OVEBTi,type,
    list_u6098035379799741383_VEBTi: list_VEBT_VEBTi > nat > vEBT_VEBTi > list_VEBT_VEBTi ).

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

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

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

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

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

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

thf(sy_c_List_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
    nth_option_nat: list_option_nat > nat > option_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____BuildupMemImp__OVEBTi_M_Eo_J,type,
    nth_Pr3306050735993963089EBTi_o: list_P8833571063612306856EBTi_o > nat > produc5014006835512566296EBTi_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Real__Oreal_J,type,
    nth_Pr3433448822664029129i_real: list_P8536626330812492744i_real > nat > produc6680258955013199682i_real ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr6329974346453275474_VEBTi: list_P785718909624839377_VEBTi > nat > produc3777764054643897931_VEBTi ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr8725177398587324397T_VEBT: list_P5988454224134618948T_VEBT > nat > produc2810682830582626868T_VEBT ).

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__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__Real__Oreal_J,type,
    nth_Pr6842391030413306568T_real: list_P2623026923184700063T_real > nat > produc5170161368751668367T_real ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr316670251186196177_VEBTi: list_P735349106241217576_VEBTi > nat > produc3625547720036274456_VEBTi ).

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

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

thf(sy_c_List_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
    nth_VEBT_VEBTi: list_VEBT_VEBTi > nat > vEBT_VEBTi ).

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__Real__Oreal,type,
    product_o_real: list_o > list_real > list_P5232166724548748803o_real ).

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__Real__Oreal,type,
    product_nat_real: list_nat > list_real > list_P3644420460460130531t_real ).

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__Real__Oreal_001_Eo,type,
    product_real_o: list_real > list_o > list_P3595434254542482545real_o ).

thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__Int__Oint,type,
    product_real_int: list_real > list_int > list_P4344331454722006975al_int ).

thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__Nat__Onat,type,
    product_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).

thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__Real__Oreal,type,
    product_real_real: list_real > list_real > list_P8689742595348180415l_real ).

thf(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    product_VEBT_VEBTi_o: list_VEBT_VEBTi > list_o > list_P8833571063612306856EBTi_o ).

thf(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    produc5476717833281694120i_real: list_VEBT_VEBTi > list_real > list_P8536626330812492744i_real ).

thf(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc194614972289024177_VEBTi: list_VEBT_VEBTi > list_VEBT_VEBTi > list_P785718909624839377_VEBTi ).

thf(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    produc1285381384045549624T_VEBT: list_VEBT_VEBTi > list_VEBT_VEBT > list_P5988454224134618948T_VEBT ).

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

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

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc4908677263432625371T_real: list_VEBT_VEBT > list_real > list_P2623026923184700063T_real ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc316462671093861988_VEBTi: list_VEBT_VEBT > list_VEBT_VEBTi > list_P735349106241217576_VEBTi ).

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

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

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

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

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

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

thf(sy_c_List_Oreplicate_001t__VEBT____BuildupMemImp__OVEBTi,type,
    replicate_VEBT_VEBTi: nat > vEBT_VEBTi > list_VEBT_VEBTi ).

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

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_Onat_Ocase__nat_001_Eo,type,
    case_nat_o: $o > ( nat > $o ) > nat > $o ).

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_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_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
    size_size_list_o: list_o > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
    size_size_list_int: list_int > 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__Option__Ooption_It__Nat__Onat_J_J,type,
    size_s6086282163384603972on_nat: list_option_nat > 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,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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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,
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thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__Uint32__Ouint32,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Nat__Bijection_Oset__decode,type,
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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,
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thf(sy_c_Num_Oinc,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
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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,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
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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_Onum_OBit0,type,
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thf(sy_c_Num_Onum_OBit1,type,
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thf(sy_c_Num_Onum_OOne,type,
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thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Num_Onum_Osize__num,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
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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,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: num > real ).

thf(sy_c_Num_Opow,type,
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thf(sy_c_Num_Opred__numeral,type,
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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__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    none_P533106815845188193et_nat: option936205604648967762et_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
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thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
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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__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,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_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
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thf(sy_c_Option_Ooption_OSome_001t__Real__Oreal,type,
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thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
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thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
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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,
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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_001t__Assertions__Oassn,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Code____Numeral__Ointeger_J,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,
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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__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,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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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,
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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,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Rat__Orat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
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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,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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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__Rat__Orat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
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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_001t__String__Ochar,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__String__Ochar,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
    ord_max_Code_integer: code_integer > code_integer > code_integer ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
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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,
    ord_max_rat: rat > rat > rat ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).

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

thf(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Numeral____Type__Onum0_J,type,
    top_to3689904424835650196l_num0: set_Numeral_num0 ).

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,
    top_top_set_char: set_char ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Oliteral_J,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Assertions__Oassn,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,
    power_power_int: int > nat > int ).

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

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

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

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Product__Type_OPair_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,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__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,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_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Pure_Otype_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_N8448461349408098053l_num1: itself8794530163899892676l_num1 ).

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

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

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,
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thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
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thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
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thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
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thf(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_Refine__Imp__Hol_Orefines_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Assertions__Oassn,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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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,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
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thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
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thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osummable_001t__Int__Oint,type,
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thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
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thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
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thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osums_001t__Int__Oint,type,
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thf(sy_c_Series_Osums_001t__Nat__Onat,type,
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thf(sy_c_Series_Osums_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_OCollect_001t__Int__Oint,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Real__Oreal_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
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thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Set_Oimage_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
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thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Set_Oinsert_001_Eo,type,
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thf(sy_c_Set_Oinsert_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
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thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
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thf(sy_c_Set_Oinsert_001t__Num__Onum,type,
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thf(sy_c_Set_Oinsert_001t__Rat__Orat,type,
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thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
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thf(sy_c_Set_Oinsert_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Num__Onum,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Rat__Orat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).

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

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

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Code____Numeral__Ointeger,type,
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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__Code____Numeral__Ointeger,type,
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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,
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thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Code____Numeral__Ointeger,type,
    set_or5754767410780653050nteger: code_integer > set_Code_integer ).

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_Signed__Division_Osigned__division__class_Osigned__divide_001t__Int__Oint,type,
    signed6714573509424544716de_int: int > int > int ).

thf(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo_001t__Int__Oint,type,
    signed6292675348222524329lo_int: int > int > int ).

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_Ochar_Osize__char,type,
    size_char: char > nat ).

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

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

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

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Nat__Onat_J,type,
    time_TBOUND_list_nat: heap_T290393402774840812st_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    time_T3808005469503390304on_nat: heap_T5317711798761887292on_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_T8149879359713347829_VEBTi: heap_T4980287057938770641_VEBTi > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__Nat__Onat,type,
    time_TBOUND_nat: heap_Time_Heap_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_T8353473612707095248on_nat: heap_T2636463487746394924on_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_T5737551269749752165_VEBTi: heap_T8145700208782473153_VEBTi > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001t__Nat__Onat,type,
    time_htt_nat: assn > heap_Time_Heap_nat > ( nat > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_htt_option_nat: assn > heap_T2636463487746394924on_nat > ( option_nat > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_htt_VEBT_VEBTi: assn > heap_T8145700208782473153_VEBTi > ( vEBT_VEBTi > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Otime_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_t3534373299052942712_VEBTi: heap_T4980287057938770641_VEBTi > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_time_VEBT_VEBTi: heap_T8145700208782473153_VEBTi > heap_e7401611519738050253t_unit > nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_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__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_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____2,type,
    type_l31302759751748492nite_2: itself_finite_2 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____3,type,
    type_l31302759751748493nite_3: itself_finite_3 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_l796852477590012082l_num1: itself8794530163899892676l_num1 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum0,type,
    type_l4264026598287037464l_num0: itself_Numeral_num0 > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: nat > nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: nat > int ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: nat > int ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: nat > nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: nat > nat > heap_Time_Heap_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: nat > nat > heap_Time_Heap_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001_Eo,type,
    vEBT_V2326993469660664182atei_o: nat > heap_Time_Heap_o > heap_T844314716496656296list_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Nat__Onat,type,
    vEBT_V7726092123322077554ei_nat: nat > heap_Time_Heap_nat > heap_T290393402774840812st_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_V792416675989592002on_nat: nat > heap_T2636463487746394924on_nat > heap_T5317711798761887292on_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_V1859673955506687831_VEBTi: nat > heap_T8145700208782473153_VEBTi > heap_T4980287057938770641_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: vEBT_VEBT > vEBT_VEBTi > nat > heap_Time_Heap_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_OLeafi,type,
    vEBT_Leafi: $o > $o > vEBT_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_ONodei,type,
    vEBT_Nodei: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > vEBT_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    vEBT_c1335663792808957512ap_nat: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_nat ) > ( $o > $o > heap_Time_Heap_nat ) > vEBT_VEBTi > heap_Time_Heap_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    vEBT_c6250501799366334488on_nat: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat ) > ( $o > $o > heap_T2636463487746394924on_nat ) > vEBT_VEBTi > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    vEBT_c6028912655521741485_VEBTi: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi ) > ( $o > $o > heap_T8145700208782473153_VEBTi ) > vEBT_VEBTi > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Nat__Onat,type,
    vEBT_case_VEBTi_nat: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat ) > ( $o > $o > nat ) > vEBT_VEBTi > nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
    vEBT_size_VEBTi: vEBT_VEBTi > nat ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: vEBT_VEBT > vEBT_VEBTi > assn ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
    vEBT_v8524038756793281170aw_rel: produc3625547720036274456_VEBTi > produc3625547720036274456_VEBTi > $o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__maxti__rel,type,
    vEBT_vebt_maxti_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__minti__rel,type,
    vEBT_vebt_minti_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).

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

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

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

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

thf(sy_c_VEBT__Definitions_OVEBT__internal_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_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__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

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

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

thf(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $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__List__Assn_OlistI__assn_001_Eo_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6286945158656146733_VEBTi: set_nat > ( $o > vEBT_VEBTi > assn ) > list_o > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1319876754960170684T_VEBT: set_nat > ( $o > vEBT_VEBT > assn ) > list_o > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2018189785592951398T_VEBT: set_nat > ( int > vEBT_VEBT > assn ) > list_int > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L7489483478785760935_VEBTi: set_nat > ( nat > vEBT_VEBTi > assn ) > list_nat > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8511957252848910786T_VEBT: set_nat > ( nat > vEBT_VEBT > assn ) > list_nat > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__Nat__Onat,type,
    vEBT_L234762979517870878al_nat: set_nat > ( real > nat > assn ) > list_real > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L7851252805511451907_VEBTi: set_nat > ( real > vEBT_VEBTi > assn ) > list_real > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3095048238742455910T_VEBT: set_nat > ( real > vEBT_VEBT > assn ) > list_real > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    vEBT_L3328983362619735041EBTi_o: set_nat > ( vEBT_VEBTi > $o > assn ) > list_VEBT_VEBTi > list_o > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    vEBT_L2806540629473551875Ti_int: set_nat > ( vEBT_VEBTi > int > assn ) > list_VEBT_VEBTi > list_int > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    vEBT_L2809031099982602151Ti_nat: set_nat > ( vEBT_VEBTi > nat > assn ) > list_VEBT_VEBTi > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    vEBT_L7728200936804140803i_real: set_nat > ( vEBT_VEBTi > real > assn ) > list_VEBT_VEBTi > list_real > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L886525131989349516_VEBTi: set_nat > ( vEBT_VEBTi > vEBT_VEBTi > assn ) > list_VEBT_VEBTi > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2497118539674116125T_VEBT: set_nat > ( vEBT_VEBTi > vEBT_VEBT > assn ) > list_VEBT_VEBTi > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    vEBT_L7058566406413635588VEBT_o: set_nat > ( vEBT_VEBT > $o > assn ) > list_VEBT_VEBT > list_o > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    vEBT_L8648204552663881920BT_int: set_nat > ( vEBT_VEBT > int > assn ) > list_VEBT_VEBT > list_int > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8650695023172932196BT_nat: set_nat > ( vEBT_VEBT > nat > assn ) > list_VEBT_VEBT > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    vEBT_L4281036506115550016T_real: set_nat > ( vEBT_VEBT > real > assn ) > list_VEBT_VEBT > list_real > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1528199826722428489_VEBTi: set_nat > ( vEBT_VEBT > vEBT_VEBTi > assn ) > list_VEBT_VEBT > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3204528365124325536T_VEBT: set_nat > ( vEBT_VEBT > vEBT_VEBT > assn ) > list_VEBT_VEBT > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001_Eo,type,
    vEBT_L7363604446928714179sn_o_o: ( $o > $o > assn ) > list_o > list_o > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Nat__Onat,type,
    vEBT_L4785011123346445925_o_nat: ( $o > nat > assn ) > list_o > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Real__Oreal,type,
    vEBT_L4725278957065240257o_real: ( $o > real > assn ) > list_o > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    vEBT_L4260503343685368993omplex: ( complex > complex > assn ) > list_complex > list_complex > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Int__Oint,type,
    vEBT_L134985006839036959ex_int: ( complex > int > assn ) > list_complex > list_int > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    vEBT_L137475477348087235ex_nat: ( complex > nat > assn ) > list_complex > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    vEBT_L2479436891206192927x_real: ( complex > real > assn ) > list_complex > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8524933119956041985T_VEBT: ( complex > vEBT_VEBT > assn ) > list_complex > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001_Eo,type,
    vEBT_L6066640139021943271_int_o: ( int > $o > assn ) > list_int > list_o > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Real__Oreal,type,
    vEBT_L8288995350762215837t_real: ( int > real > assn ) > list_int > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001_Eo,type,
    vEBT_L7887682484454631235_nat_o: ( nat > $o > assn ) > list_nat > list_o > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Real__Oreal,type,
    vEBT_L6102073776069194049t_real: ( nat > real > assn ) > list_nat > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001_Eo,type,
    vEBT_L6234343332106409831real_o: ( real > $o > assn ) > list_real > list_o > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Nat__Onat,type,
    vEBT_L1446010312343316929al_nat: ( real > nat > assn ) > list_real > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Real__Oreal,type,
    vEBT_L1930518968523514909l_real: ( real > real > assn ) > list_real > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L9060850011106065574_VEBTi: ( real > vEBT_VEBTi > assn ) > list_real > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L4595930785310033027T_VEBT: ( real > vEBT_VEBT > assn ) > list_real > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    vEBT_L8927591528087875366Ti_int: ( vEBT_VEBTi > int > assn ) > list_VEBT_VEBTi > list_int > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    vEBT_L8930081998596925642Ti_nat: ( vEBT_VEBTi > nat > assn ) > list_VEBT_VEBTi > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1891944875198410415_VEBTi: ( vEBT_VEBTi > vEBT_VEBTi > assn ) > list_VEBT_VEBTi > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L7265847600308530106T_VEBT: ( vEBT_VEBTi > vEBT_VEBT > assn ) > list_VEBT_VEBTi > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    vEBT_L7489408758114837031VEBT_o: ( vEBT_VEBT > $o > assn ) > list_VEBT_VEBT > list_o > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
    vEBT_L2162147798726695391omplex: ( vEBT_VEBT > complex > assn ) > list_VEBT_VEBT > list_complex > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    vEBT_L8294436054247626077BT_int: ( vEBT_VEBT > int > assn ) > list_VEBT_VEBT > list_int > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8296926524756676353BT_nat: ( vEBT_VEBT > nat > assn ) > list_VEBT_VEBT > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_L8010285020845282001on_nat: ( vEBT_VEBT > option_nat > assn ) > list_VEBT_VEBT > list_option_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    vEBT_L5781919052683127133T_real: ( vEBT_VEBT > real > assn ) > list_VEBT_VEBT > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6296928887356842470_VEBTi: ( vEBT_VEBT > vEBT_VEBTi > assn ) > list_VEBT_VEBT > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1279224858307276611T_VEBT: ( vEBT_VEBT > vEBT_VEBT > assn ) > list_VEBT_VEBT > list_VEBT_VEBT > assn ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: nat > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: nat > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: nat > nat > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: nat > nat > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: vEBT_VEBT > real ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: 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__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__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__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    accp_P7675410724331315407_VEBTi: ( produc3625547720036274456_VEBTi > produc3625547720036274456_VEBTi > $o ) > produc3625547720036274456_VEBTi > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__VEBT____BuildupMemImp__OVEBTi,type,
    accp_VEBT_VEBTi: ( vEBT_VEBTi > vEBT_VEBTi > $o ) > vEBT_VEBTi > $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_fChoice_001t__Real__Oreal,type,
    fChoice_real: ( real > $o ) > real ).

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

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

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

thf(sy_c_member_001t__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__Real__Oreal_J,type,
    member_list_real: list_real > set_list_real > $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__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member6260224972018164377et_nat: produc3658429121746597890et_nat > set_Pr3948176798113811640et_nat > $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____BuildupMemImp__OVEBTi,type,
    member_VEBT_VEBTi: vEBT_VEBTi > set_VEBT_VEBTi > $o ).

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

thf(sy_v_ma____,type,
    ma: nat ).

thf(sy_v_mi____,type,
    mi: nat ).

thf(sy_v_summary____,type,
    summary: vEBT_VEBT ).

thf(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

thf(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_tree__is__103_058ATP,type,
    tree_is_103_ATP: list_VEBT_VEBTi ).

thf(sy_v_tree__is______,type,
    tree_is: list_VEBT_VEBTi ).

thf(sy_v_va____,type,
    va: nat ).

thf(sy_v_x13______,type,
    x13: array_VEBT_VEBTi ).

thf(sy_v_x14______,type,
    x14: vEBT_VEBTi ).

thf(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (10205)
thf(fact_0_VEBTi_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,Y11: option4927543243414619207at_nat,Y12: nat,Y13: array_VEBT_VEBTi,Y14: vEBT_VEBTi] :
      ( ( ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 )
        = ( vEBT_Nodei @ Y11 @ Y12 @ Y13 @ Y14 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBTi.inject(1)
thf(fact_1_assnle,axiom,
    ! [TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] : ( entails @ ( times_times_assn @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ).

% assnle
thf(fact_2_zero__less__power2,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
thf(fact_3_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_4_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_5_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_6_div2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div2_Suc_Suc
thf(fact_7_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_8_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_9_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_10_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_11_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_12_zero__eq__power2,axiom,
    ! [A: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% zero_eq_power2
thf(fact_13_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_14_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_15_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_16_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_17_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_18_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_19_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_20_power__eq__0__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ( power_8256067586552552935nteger @ A @ N )
        = zero_z3403309356797280102nteger )
      = ( ( A = zero_z3403309356797280102nteger )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_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_29_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_30_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_31_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_32_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_33_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_34_odd__power__less__zero,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_z3403309356797280102nteger ) ) ).

% odd_power_less_zero
thf(fact_35_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = ( divide_divide_nat @ M @ N ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_36_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numera6690914467698888265omplex @ M )
        = ( numera6690914467698888265omplex @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_37_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_38_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_rat @ M )
        = ( numeral_numeral_rat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_39_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_40_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_41_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(78)
thf(fact_42_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).

% semiring_norm(13)
thf(fact_43_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(87)
thf(fact_44_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_45_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_46_semiring__norm_I12_J,axiom,
    ! [N: num] :
      ( ( times_times_num @ one @ N )
      = N ) ).

% semiring_norm(12)
thf(fact_47_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_48_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_49_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_50_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_51_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_52_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_53_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_54_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_55_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_56_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_57_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_58_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_59_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_60_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_61_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_62_semiring__norm_I76_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).

% semiring_norm(76)
thf(fact_63_num__double,axiom,
    ! [N: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N )
      = ( bit0 @ N ) ) ).

% num_double
thf(fact_64_semiring__norm_I83_J,axiom,
    ! [N: num] :
      ( one
     != ( bit0 @ N ) ) ).

% semiring_norm(83)
thf(fact_65_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_66_power__mult__numeral,axiom,
    ! [A: nat,M: num,N: num] :
      ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_67_power__mult__numeral,axiom,
    ! [A: real,M: num,N: num] :
      ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_68_power__mult__numeral,axiom,
    ! [A: int,M: num,N: num] :
      ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_69_power__mult__numeral,axiom,
    ! [A: complex,M: num,N: num] :
      ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_70_power__mult__numeral,axiom,
    ! [A: code_integer,M: num,N: num] :
      ( ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_71_power__mult__numeral,axiom,
    ! [A: rat,M: num,N: num] :
      ( ( power_power_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_72_div__mult__mult1__if,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ( C = zero_z3403309356797280102nteger )
       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
          = zero_z3403309356797280102nteger ) )
      & ( ( C != zero_z3403309356797280102nteger )
       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_73_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_74_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_75_div__mult__mult2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_76_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_77_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_78_div__mult__mult1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_79_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_80_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_81_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_82_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_83_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_84_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_85_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( suc @ N ) )
      = zero_z3403309356797280102nteger ) ).

% power_0_Suc
thf(fact_86_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_87_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_88_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_89_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_90_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_91_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( numeral_numeral_nat @ K ) )
      = zero_z3403309356797280102nteger ) ).

% power_zero_numeral
thf(fact_92_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_93_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_94_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_95_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_96_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_97_power__Suc0__right,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_98_power__Suc0__right,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_99_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_100_div__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_101_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( ( power_power_nat @ X @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_102_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_103_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_104_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_105_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
      = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_106_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
      = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_107_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_108_mem__Collect__eq,axiom,
    ! [A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( member_VEBT_VEBT @ A @ ( collect_VEBT_VEBT @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_109_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_111_mem__Collect__eq,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_112_mem__Collect__eq,axiom,
    ! [A: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_113_Collect__mem__eq,axiom,
    ! [A2: set_real] :
      ( ( collect_real
        @ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ^ [X2: vEBT_VEBT] : ( member_VEBT_VEBT @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_116_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_117_Collect__mem__eq,axiom,
    ! [A2: set_complex] :
      ( ( collect_complex
        @ ^ [X2: complex] : ( member_complex @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_118_Collect__mem__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_119_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_120_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_121_Collect__cong,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_complex @ P )
        = ( collect_complex @ Q ) ) ) ).

% Collect_cong
thf(fact_122_Collect__cong,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec213857154873943460nt_int @ P )
        = ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_cong
thf(fact_123_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_124_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_125_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_126_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_127_less__numeral__extra_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_numeral_extra(3)
thf(fact_128_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( numera6620942414471956472nteger @ N ) ) ).

% zero_neq_numeral
thf(fact_129_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N ) ) ).

% zero_neq_numeral
thf(fact_130_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N ) ) ).

% zero_neq_numeral
thf(fact_131_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N ) ) ).

% zero_neq_numeral
thf(fact_132_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N ) ) ).

% zero_neq_numeral
thf(fact_133_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N ) ) ).

% zero_neq_numeral
thf(fact_134_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N )
       != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_135_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N )
       != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_136_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N )
       != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_137_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( ( power_power_complex @ A @ N )
       != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_138_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: code_integer,N: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ A @ N )
       != zero_z3403309356797280102nteger ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_139_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( ( power_power_rat @ A @ N )
       != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_140_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_141_power__commuting__commutes,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = ( times_3573771949741848930nteger @ Y @ X ) )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ Y )
        = ( times_3573771949741848930nteger @ Y @ ( power_8256067586552552935nteger @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_142_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_143_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_144_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_145_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_146_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_147_power__mult__distrib,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( times_3573771949741848930nteger @ A @ B ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_148_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_149_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_150_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_151_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_152_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_153_power__commutes,axiom,
    ! [A: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N ) @ A )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_commutes
thf(fact_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_162_power__mult,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_163_power__mult,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_164_power__mult,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_165_power__mult,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_166_power__mult,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( times_times_nat @ M @ N ) )
      = ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_167_power__mult,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( power_power_rat @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_rat @ ( power_power_rat @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_168_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).

% div_mult2_eq
thf(fact_169_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).

% not_numeral_less_zero
thf(fact_170_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_171_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_172_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_173_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_174_zero__less__numeral,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).

% zero_less_numeral
thf(fact_175_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_less_numeral
thf(fact_176_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_less_numeral
thf(fact_177_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_less_numeral
thf(fact_178_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_less_numeral
thf(fact_179_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_180_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_181_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_182_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_183_zero__less__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% zero_less_power
thf(fact_184_mult__numeral__1__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_185_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_186_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_187_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_188_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_189_mult__numeral__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_190_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_191_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_192_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_193_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_194_divide__numeral__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_195_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_196_divide__numeral__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_197_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_198_power__Suc2,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_199_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_200_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_201_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_202_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_203_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_204_power__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_Suc
thf(fact_205_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_206_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_207_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_208_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_209_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N )
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_210_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_211_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N ) )
        = ( M = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_212_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_213_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
        = zero_z3403309356797280102nteger ) ) ).

% zero_power
thf(fact_227_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_228_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_229_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).

% power_gt_expt
thf(fact_230_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
        = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_231_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( divide_divide_nat @ M @ N ) ) ) ).

% nat_mult_div_cancel1
thf(fact_232_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_233_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_234_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_235_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_236_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_237_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_238_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_239_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_240_zero__power2,axiom,
    ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% zero_power2
thf(fact_241_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_242_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_243_power2__eq__square,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% power2_eq_square
thf(fact_244_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_245_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_246_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_247_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_248_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_249_power4__eq__xxxx,axiom,
    ! [X: code_integer] :
      ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_250_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_251_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_252_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_253_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_254_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_255_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_256_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_257_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_258_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_259_power__even__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_260_power__even__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_261_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_262_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_263_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_264_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_265_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_266_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_267_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_268_power2__less__0,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger ) ).

% power2_less_0
thf(fact_269_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_270_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_271_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_272_power__odd__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_273_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_274_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_275_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_276_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_277_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_278_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_279_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_280_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% mult_less_cancel2
thf(fact_281_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_282_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_283_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_284_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_285_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_286_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_287_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_288_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_289_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_290_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_291_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_292_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_293_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_294_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_295_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_296_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_297_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_298_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_299_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_300_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_301_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_302_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_303_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_304_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_305_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_306_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_307_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_308_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_309_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% nat.inject
thf(fact_310_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_311_mult__zero__left,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_312_mult__zero__left,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mult_zero_left
thf(fact_313_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_314_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_315_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_316_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_317_mult__zero__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_318_mult__zero__right,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% mult_zero_right
thf(fact_319_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_320_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_321_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_322_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_323_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_324_mult__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% mult_eq_0_iff
thf(fact_325_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_326_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_327_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_328_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_329_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_330_mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ C @ A )
        = ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_331_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_332_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_333_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_334_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_335_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_336_mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ C )
        = ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_337_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_338_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_339_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_340_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_341_div__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% div_0
thf(fact_342_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_343_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_344_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_345_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_346_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_347_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_348_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_349_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_350_div__by__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% div_by_0
thf(fact_351_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_352_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_353_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_354_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_355_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_356_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_357_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_358_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_359_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_360_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_361_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_362_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_363_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_364_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_365_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_366_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_367_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_368_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_369_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_370_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_371_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_372_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_373_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_374_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_375_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_376_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_377_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_378_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_379_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_380_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_381_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_382_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_383_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_384_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_385_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_386_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N @ K ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_387_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_388_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_389_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_390_linordered__field__no__lb,axiom,
    ! [X4: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X4 ) ).

% linordered_field_no_lb
thf(fact_391_linordered__field__no__lb,axiom,
    ! [X4: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X4 ) ).

% linordered_field_no_lb
thf(fact_392_linordered__field__no__ub,axiom,
    ! [X4: real] :
    ? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_393_linordered__field__no__ub,axiom,
    ! [X4: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X4 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_394_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_395_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_396_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_397_linorder__neqE__linordered__idom,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( X != Y )
     => ( ~ ( ord_le6747313008572928689nteger @ X @ Y )
       => ( ord_le6747313008572928689nteger @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_398_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_399_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_400_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_401_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_402_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_403_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_404_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_405_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
             => ( P @ M2 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_406_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_407_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_408_size__neq__size__imp__neq,axiom,
    ! [X: list_real,Y: list_real] :
      ( ( ( size_size_list_real @ X )
       != ( size_size_list_real @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_409_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_410_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_411_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_412_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_413_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_414_mult__not__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ B )
       != zero_z3403309356797280102nteger )
     => ( ( A != zero_z3403309356797280102nteger )
        & ( B != zero_z3403309356797280102nteger ) ) ) ).

% mult_not_zero
thf(fact_415_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_416_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_417_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_418_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_419_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_420_divisors__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% divisors_zero
thf(fact_421_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_422_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_423_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_424_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_425_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_426_no__zero__divisors,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( B != zero_z3403309356797280102nteger )
       => ( ( times_3573771949741848930nteger @ A @ B )
         != zero_z3403309356797280102nteger ) ) ) ).

% no_zero_divisors
thf(fact_427_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_428_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_429_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_430_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_431_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_432_mult__left__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( ( times_3573771949741848930nteger @ C @ A )
          = ( times_3573771949741848930nteger @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_433_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_434_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_435_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_436_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_437_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_438_mult__right__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( ( times_3573771949741848930nteger @ A @ C )
          = ( times_3573771949741848930nteger @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_439_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_440_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_441_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_442_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_443_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_444_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_445_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_446_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_447_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_448_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_449_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_450_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_451_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_452_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_453_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_454_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_455_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_456_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_457_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_458_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X3: nat,Y3: nat] :
              ( ( P @ X3 @ Y3 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_459_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_460_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_461_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_462_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_463_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M3: nat] :
          ( N
          = ( suc @ M3 ) ) ) ).

% not0_implies_Suc
thf(fact_464_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J: nat] :
              ( ( ord_less_nat @ I @ J )
             => ( K
               != ( suc @ J ) ) ) ) ) ).

% Nat.lessE
thf(fact_465_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_466_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J: nat] :
            ( ( ord_less_nat @ I @ J )
           => ( K
             != ( suc @ J ) ) ) ) ).

% Suc_lessE
thf(fact_467_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_468_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_469_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_470_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
            & ( P @ I2 ) ) )
      = ( ( P @ N )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
            & ( P @ I2 ) ) ) ) ).

% Ex_less_Suc
thf(fact_471_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_472_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_473_Nat_OAll__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
           => ( P @ I2 ) ) )
      = ( ( P @ N )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
           => ( P @ I2 ) ) ) ) ).

% Nat.All_less_Suc
thf(fact_474_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M4: nat] :
            ( ( M
              = ( suc @ M4 ) )
            & ( ord_less_nat @ N @ M4 ) ) ) ) ).

% Suc_less_eq2
thf(fact_475_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_476_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_477_less__trans__Suc,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( ord_less_nat @ J2 @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_478_less__Suc__induct,axiom,
    ! [I: nat,J2: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J: nat,K2: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( ord_less_nat @ J @ K2 )
               => ( ( P @ I3 @ J )
                 => ( ( P @ J @ K2 )
                   => ( P @ I3 @ K2 ) ) ) ) )
         => ( P @ I @ J2 ) ) ) ) ).

% less_Suc_induct
thf(fact_479_strict__inc__induct,axiom,
    ! [I: nat,J2: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ! [I3: nat] :
            ( ( J2
              = ( suc @ I3 ) )
           => ( P @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( P @ ( suc @ I3 ) )
               => ( P @ I3 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_480_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_481_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_482_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_483_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_484_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_485_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_486_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_487_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N2 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_488_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( M = N ) ) ).

% Suc_mult_cancel1
thf(fact_489_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_490_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_491_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_492_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_493_mult__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_494_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_495_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_496_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_497_not__square__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ A ) @ zero_z3403309356797280102nteger ) ).

% not_square_less_zero
thf(fact_498_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_499_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_500_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_501_mult__less__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_502_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_503_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_504_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_505_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_506_mult__neg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_neg_pos
thf(fact_507_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_508_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_509_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_510_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_511_mult__pos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_pos_neg
thf(fact_512_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_513_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_514_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_515_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_516_mult__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_517_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_518_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_519_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_520_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_521_mult__pos__neg2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ B @ A ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_pos_neg2
thf(fact_522_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_523_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_524_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_525_zero__less__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) )
        | ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).

% zero_less_mult_iff
thf(fact_526_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_527_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_528_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_529_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_530_zero__less__mult__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_531_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_532_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_533_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_534_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_535_zero__less__mult__pos2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ B @ A ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_536_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_537_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_538_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_539_mult__less__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le6747313008572928689nteger @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_540_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_541_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_542_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_543_mult__less__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_544_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_545_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_546_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_547_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_548_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_549_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_550_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_551_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_552_mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_553_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_554_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_555_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_556_mult__less__cancel__left__disj,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
          & ( ord_le6747313008572928689nteger @ A @ B ) )
        | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_557_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_558_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_559_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_560_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_561_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_562_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_563_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_564_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_565_mult__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_566_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_567_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_568_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_569_mult__less__cancel__right__disj,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
          & ( ord_le6747313008572928689nteger @ A @ B ) )
        | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_570_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_571_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_572_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_573_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_574_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_575_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_576_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_577_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_578_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_579_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_580_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_581_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_582_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_583_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_584_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_585_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_586_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_587_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_588_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_589_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_590_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_591_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_592_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_593_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_594_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_595_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_596_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_597_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_598_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_599_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_600_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_601_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_602_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_603_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_604_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_605_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_606_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_607_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_608_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_609_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_610_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_611_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_612_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_613_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_614_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ord_less_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_615_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ord_less_rat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_616_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ord_less_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_617_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_618_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_619_lift__Suc__mono__less,axiom,
    ! [F: nat > code_integer,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_le6747313008572928689nteger @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ord_le6747313008572928689nteger @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_620_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_621_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_622_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_623_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_624_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_625_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > code_integer,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_le6747313008572928689nteger @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_le6747313008572928689nteger @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_626_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
            & ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
            & ( P @ ( suc @ I2 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_627_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M5: nat] :
            ( N
            = ( suc @ M5 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_628_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_629_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M3: nat] :
          ( N
          = ( suc @ M3 ) ) ) ).

% gr0_implies_Suc
thf(fact_630_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_631_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_mult_less_cancel1
thf(fact_632_mult__less__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_633_mult__less__mono2,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).

% mult_less_mono2
thf(fact_634_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_635_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_636_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_637_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_638_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_639_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_640_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_641_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_642_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_643_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_644_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_645_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_646_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_647_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_648_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_649_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_650_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_651_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_652_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_653_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_654_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).

% one_less_mult
thf(fact_655_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_656_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_657_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_658_listI__assn__conv_H,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,A2: vEBT_VEBT > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi,F2: assn] :
      ( ( N
        = ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) @ A2 @ Xs @ Xsi ) @ F2 )
        = ( times_times_assn @ ( vEBT_L6296928887356842470_VEBTi @ A2 @ Xs @ Xsi ) @ F2 ) ) ) ).

% listI_assn_conv'
thf(fact_659_listI__assn__conv,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,A2: vEBT_VEBT > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ( N
        = ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( vEBT_L1528199826722428489_VEBTi @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) @ A2 @ Xs @ Xsi )
        = ( vEBT_L6296928887356842470_VEBTi @ A2 @ Xs @ Xsi ) ) ) ).

% listI_assn_conv
thf(fact_660_list__assn__conv__idx,axiom,
    ( vEBT_L6296928887356842470_VEBTi
    = ( ^ [A3: vEBT_VEBT > vEBT_VEBTi > assn,Xs2: list_VEBT_VEBT] : ( vEBT_L1528199826722428489_VEBTi @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ A3 @ Xs2 ) ) ) ).

% list_assn_conv_idx
thf(fact_661_n__less__equal__power__2,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% n_less_equal_power_2
thf(fact_662_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(2)
thf(fact_663_less__option__Some,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_option_real @ ( some_real @ X ) @ ( some_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% less_option_Some
thf(fact_664_less__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_option_rat @ ( some_rat @ X ) @ ( some_rat @ Y ) )
      = ( ord_less_rat @ X @ Y ) ) ).

% less_option_Some
thf(fact_665_less__option__Some,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_option_num @ ( some_num @ X ) @ ( some_num @ Y ) )
      = ( ord_less_num @ X @ Y ) ) ).

% less_option_Some
thf(fact_666_less__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_option_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_option_Some
thf(fact_667_less__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_option_int @ ( some_int @ X ) @ ( some_int @ Y ) )
      = ( ord_less_int @ X @ Y ) ) ).

% less_option_Some
thf(fact_668_less__option__Some,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le7113747843092208513nteger @ ( some_Code_integer @ X ) @ ( some_Code_integer @ Y ) )
      = ( ord_le6747313008572928689nteger @ X @ Y ) ) ).

% less_option_Some
thf(fact_669_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_double_not_eq_double
thf(fact_670_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% double_not_eq_Suc_double
thf(fact_671_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_672_bits__div__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_div_0
thf(fact_673_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_674_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_675_bits__div__by__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% bits_div_by_0
thf(fact_676_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_677_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_678_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_679_i0__less,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
      = ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_680_half__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% half_negative_int_iff
thf(fact_681_not__iless0,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_682_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M2 @ N2 )
             => ( P @ M2 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% enat_less_induct
thf(fact_683_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).

% enat_0_less_mult_iff
thf(fact_684_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R: real] :
          ( ( ord_less_real @ zero_zero_real @ R )
          & ( ( power_power_real @ R @ ( suc @ N ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_685_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_686_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X3: real] :
            ( ( ord_less_real @ zero_zero_real @ X3 )
            & ( ( power_power_real @ X3 @ N )
              = A )
            & ! [Y4: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y4 )
                  & ( ( power_power_real @ Y4 @ N )
                    = A ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_687_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R: real] :
            ( ( ord_less_real @ zero_zero_real @ R )
            & ( ( power_power_real @ R @ N )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_688_list__assn__mono,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > assn,P2: vEBT_VEBT > vEBT_VEBTi > assn,L: list_VEBT_VEBT,L2: list_VEBT_VEBTi] :
      ( ! [X3: vEBT_VEBT,X5: vEBT_VEBTi] : ( entails @ ( P @ X3 @ X5 ) @ ( P2 @ X3 @ X5 ) )
     => ( entails @ ( vEBT_L6296928887356842470_VEBTi @ P @ L @ L2 ) @ ( vEBT_L6296928887356842470_VEBTi @ P2 @ L @ L2 ) ) ) ).

% list_assn_mono
thf(fact_689_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_690_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_691_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_692_td__gal__lt,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( ord_less_nat @ ( divide_divide_nat @ A @ C ) @ B ) ) ) ).

% td_gal_lt
thf(fact_693_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_694_unset__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_695_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_696_T__vebt__buildupi__gq__0,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( vEBT_V441764108873111860ildupi @ N ) ) ).

% T_vebt_buildupi_gq_0
thf(fact_697_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X2: nat,N4: nat] : ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% high_def
thf(fact_698_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_nat @ M5 @ N )
           => ( P @ M5 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X2 ) ) ) ) ).

% all_nat_less_eq
thf(fact_699_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_nat @ M5 @ N )
            & ( P @ M5 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X2 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_700_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_701_option_Osize_I4_J,axiom,
    ! [X22: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_702_option_Osize_I4_J,axiom,
    ! [X22: num] :
      ( ( size_size_option_num @ ( some_num @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_703_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ X ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ X )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% Comparator_Generator.All_less_Suc
thf(fact_704_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_705_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_706_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_707_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_708_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_709_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_710_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_711_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_712_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_713_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X22 )
        = ( some_P7363390416028606310at_nat @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_714_option_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( ( some_nat @ X22 )
        = ( some_nat @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_715_option_Oinject,axiom,
    ! [X22: num,Y2: num] :
      ( ( ( some_num @ X22 )
        = ( some_num @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_716_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_717_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_718_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_719_add_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.right_neutral
thf(fact_720_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_721_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_722_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_723_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_724_add_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% add.right_neutral
thf(fact_725_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_726_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_727_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_728_double__zero__sym,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( plus_p5714425477246183910nteger @ A @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% double_zero_sym
thf(fact_729_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_730_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_731_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_732_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_733_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_734_add__cancel__left__left,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B @ A )
        = A )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_left
thf(fact_735_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_736_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_737_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_738_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_739_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_740_add__cancel__left__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = A )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_right
thf(fact_741_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_742_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_743_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_744_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_745_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_746_add__cancel__right__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_left
thf(fact_747_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_748_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_749_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_750_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_751_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_752_add__cancel__right__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_right
thf(fact_753_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_754_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_755_add__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add_0
thf(fact_756_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_757_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_758_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_759_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_760_add__0,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% add_0
thf(fact_761_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_762_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_763_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_764_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_765_add__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_766_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_767_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_768_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_769_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_770_add__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_771_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_772_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_773_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_774_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_775_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_776_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_777_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_778_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_779_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_780_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_781_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_782_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_783_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_784_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_785_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_786_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L3: nat,D: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D ) ) @ L3 ) ) ) ).

% bit_concat_def
thf(fact_787_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_788_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_789_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_790_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_791_add__less__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_less_same_cancel1
thf(fact_792_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_793_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_794_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_795_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_796_add__less__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_less_same_cancel2
thf(fact_797_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_798_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_799_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_800_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_801_less__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% less_add_same_cancel1
thf(fact_802_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_803_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_804_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_805_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_806_less__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% less_add_same_cancel2
thf(fact_807_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_808_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_809_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_810_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_811_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_812_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_813_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_814_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_815_sum__squares__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
        = zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_816_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_817_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_818_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_819_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_820_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_821_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_822_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_823_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_824_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_825_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_826_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_827_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_828_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_829_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_830_mult__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ M @ ( suc @ N ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc_right
thf(fact_831_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_832_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_833_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_834_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_835_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_836_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_837_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_838_div__mult__self3,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_839_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_840_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_841_div__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_842_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_843_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_844_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_845_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_846_add__self__div__2,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = M ) ).

% add_self_div_2
thf(fact_847_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_848_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_849_sum__power2__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_850_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_851_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_852_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ M @ N )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
thf(fact_853_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_854_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_855_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_856_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_857_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_858_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_859_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_860_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_861_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_862_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_863_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_864_ab__semigroup__add__class_Oadd_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 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_865_ab__semigroup__add__class_Oadd_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 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_866_ab__semigroup__add__class_Oadd_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 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_867_ab__semigroup__add__class_Oadd_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 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_868_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A4: real,B2: real] : ( plus_plus_real @ B2 @ A4 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_869_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A4: rat,B2: rat] : ( plus_plus_rat @ B2 @ A4 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_870_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B2: nat] : ( plus_plus_nat @ B2 @ A4 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_871_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A4: int,B2: int] : ( plus_plus_int @ B2 @ A4 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_872_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_873_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_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_group__cancel_Oadd2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B3 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_883_group__cancel_Oadd2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B3 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_884_group__cancel_Oadd2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B3 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_885_group__cancel_Oadd2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B3 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_886_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_887_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_888_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_889_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_890_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( I = J2 )
        & ( K = L ) )
     => ( ( plus_plus_real @ I @ K )
        = ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_891_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( I = J2 )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_892_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( I = J2 )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_893_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( I = J2 )
        & ( K = L ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_894_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_895_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_896_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_897_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_898_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_899_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_900_add_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.comm_neutral
thf(fact_901_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_902_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_903_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_904_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_905_add_Ocomm__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% add.comm_neutral
thf(fact_906_add_Ogroup__left__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_907_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_908_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_909_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_910_add_Ogroup__left__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_911_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J2 )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_912_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J2 )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_913_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J2 )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_914_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J2 )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_915_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: code_integer,J2: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I @ J2 )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_916_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( I = J2 )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_917_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( I = J2 )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_918_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( I = J2 )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_919_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( I = J2 )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_920_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: code_integer,J2: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I = J2 )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_921_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J2 )
        & ( K = L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_922_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J2 )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_923_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J2 )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_924_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J2 )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_925_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: code_integer,J2: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I @ J2 )
        & ( K = L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_926_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D2 )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_927_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_928_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_929_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_930_add__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_931_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_932_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_933_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_934_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_935_add__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_936_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_937_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_938_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_939_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_940_add__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_941_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_942_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_943_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_944_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_945_add__less__imp__less__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
     => ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_946_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_947_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_948_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_949_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_950_add__less__imp__less__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
     => ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_951_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_952_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_953_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_954_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_955_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_956_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_957_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_958_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_959_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_960_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_961_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_962_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_963_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_964_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_965_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_966_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_967_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_968_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_969_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_970_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_971_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_972_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_973_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_974_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_975_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_976_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A2 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_977_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_978_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_979_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_980_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_981_add__lessD1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_982_add__less__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).

% add_less_mono
thf(fact_983_not__add__less1,axiom,
    ! [I: nat,J2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).

% not_add_less1
thf(fact_984_not__add__less2,axiom,
    ! [J2: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).

% not_add_less2
thf(fact_985_add__less__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% add_less_mono1
thf(fact_986_trans__less__add1,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).

% trans_less_add1
thf(fact_987_trans__less__add2,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).

% trans_less_add2
thf(fact_988_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_989_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% add_mult_distrib
thf(fact_990_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% add_mult_distrib2
thf(fact_991_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J2: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J2 ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_992_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_993_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_994_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_995_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_996_add__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_neg_neg
thf(fact_997_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_998_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_999_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_1000_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_1001_add__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_1002_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_1003_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_1004_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_1005_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_1006_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_1007_pos__add__strict,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1008_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_1009_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_1010_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_1011_add__less__zeroD,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ X @ zero_z3403309356797280102nteger )
        | ( ord_le6747313008572928689nteger @ Y @ zero_z3403309356797280102nteger ) ) ) ).

% add_less_zeroD
thf(fact_1012_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_Bit0
thf(fact_1013_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_1014_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_1015_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_1016_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_1017_power__add,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_add
thf(fact_1018_power__add,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_add
thf(fact_1019_power__add,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).

% power_add
thf(fact_1020_power__add,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_add
thf(fact_1021_power__add,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).

% power_add
thf(fact_1022_power__add,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).

% power_add
thf(fact_1023_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1024_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1025_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q3: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).

% less_natE
thf(fact_1026_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_1027_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_1028_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N4: nat] :
        ? [K3: nat] :
          ( N4
          = ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1029_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1030_less__imp__add__positive,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J2 ) ) ) ).

% less_imp_add_positive
thf(fact_1031_mult__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc
thf(fact_1032_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    ( ( vEBT_V441764108873111860ildupi @ ( suc @ zero_zero_nat ) )
    = ( suc @ zero_zero_nat ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
thf(fact_1033_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    ( ( vEBT_V441764108873111860ildupi @ zero_zero_nat )
    = ( suc @ zero_zero_nat ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
thf(fact_1034_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_1035_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_1036_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_1037_not__sum__squares__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger ) ).

% not_sum_squares_lt_zero
thf(fact_1038_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_1039_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_1040_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_1041_sum__squares__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) )
      = ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_1042_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_1043_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_1044_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_1045_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_1046_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_1047_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_1048_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_1049_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_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_num_Osize_I5_J,axiom,
    ! [X22: num] :
      ( ( size_size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_1064_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
      = ( X = zero_zero_complex ) ) ).

% zero_reorient
thf(fact_1065_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_1066_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_1067_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_1068_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_1069_zero__reorient,axiom,
    ! [X: code_integer] :
      ( ( zero_z3403309356797280102nteger = X )
      = ( X = zero_z3403309356797280102nteger ) ) ).

% zero_reorient
thf(fact_1070_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_1071_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_1072_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_1073_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_1074_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A4: real,B2: real] : ( times_times_real @ B2 @ A4 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_1075_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A4: rat,B2: rat] : ( times_times_rat @ B2 @ A4 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_1076_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A4: nat,B2: nat] : ( times_times_nat @ B2 @ A4 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_1077_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A4: int,B2: int] : ( times_times_int @ B2 @ A4 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_1078_ab__semigroup__mult__class_Omult_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 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_1079_ab__semigroup__mult__class_Omult_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 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_1080_ab__semigroup__mult__class_Omult_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 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_1081_ab__semigroup__mult__class_Omult_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 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_1082_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_real] :
      ( ( size_size_list_real @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1083_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1084_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1085_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_int] :
      ( ( size_size_list_int @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1086_neq__if__length__neq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( ( size_size_list_real @ Xs )
       != ( size_size_list_real @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_1087_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs )
       != ( size_size_list_o @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_1088_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_1089_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
       != ( size_size_list_int @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_1090_mult__2,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2
thf(fact_1091_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_1092_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_1093_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_1094_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_1095_mult__2__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2_right
thf(fact_1096_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_1097_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_1098_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_1099_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_1100_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_1101_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_1102_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_1103_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_1104_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_1105_split__div,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) )
               => ( P @ I2 ) ) ) ) ) ) ).

% split_div
thf(fact_1106_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).

% dividend_less_div_times
thf(fact_1107_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_1108_msrevs_I1_J,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) @ N )
        = ( plus_plus_nat @ ( divide_divide_nat @ M @ N ) @ K ) ) ) ).

% msrevs(1)
thf(fact_1109_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
thf(fact_1110_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_1111_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_1112_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
thf(fact_1113_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_1114_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_1115_div__exp__eq,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_1116_div__exp__eq,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_1117_div__exp__eq,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_1118_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_1119_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_1120_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_1121_not__sum__power2__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger ) ).

% not_sum_power2_lt_zero
thf(fact_1122_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_1123_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_1124_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_1125_sum__power2__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_1126_power2__sum,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_1127_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_1128_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_1129_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_1130_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_1131_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_1132_nat__add__offset__less,axiom,
    ! [Y: nat,N: nat,X: nat,M: nat,Sz: nat] :
      ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ( Sz
            = ( plus_plus_nat @ M @ N ) )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ Y ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Sz ) ) ) ) ) ).

% nat_add_offset_less
thf(fact_1133_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_1134_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_1135_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_1136_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_1137_forall__finite_I1_J,axiom,
    ! [P: nat > $o,I4: nat] :
      ( ( ord_less_nat @ I4 @ zero_zero_nat )
     => ( P @ I4 ) ) ).

% forall_finite(1)
thf(fact_1138_length__induct,axiom,
    ! [P: list_real > $o,Xs: list_real] :
      ( ! [Xs3: list_real] :
          ( ! [Ys2: list_real] :
              ( ( ord_less_nat @ ( size_size_list_real @ Ys2 ) @ ( size_size_list_real @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_1139_length__induct,axiom,
    ! [P: list_o > $o,Xs: 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 @ Xs ) ) ).

% length_induct
thf(fact_1140_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_1141_length__induct,axiom,
    ! [P: list_int > $o,Xs: list_int] :
      ( ! [Xs3: list_int] :
          ( ! [Ys2: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_1142_forall__finite_I3_J,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ ( suc @ X ) ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ X ) )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% forall_finite(3)
thf(fact_1143_forall__finite_I2_J,axiom,
    ! [P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ zero_zero_nat ) )
           => ( P @ I2 ) ) )
      = ( P @ zero_zero_nat ) ) ).

% forall_finite(2)
thf(fact_1144_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_1145_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_1146_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_1147_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_1148_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_1149_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_1150_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_1151_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_1152_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_1153_double__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% double_eq_0_iff
thf(fact_1154_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_1155_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_1156_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y5: nat,X2: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X2 ) @ ( some_nat @ Y5 ) ) ) ) ).

% greater_shift
thf(fact_1157_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y5: nat] : ( vEBT_VEBT_less @ ( some_nat @ X2 ) @ ( some_nat @ Y5 ) ) ) ) ).

% less_shift
thf(fact_1158_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_1159_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_neg_pos_less0
thf(fact_1160_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_1161_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_1162_bit__split__inv,axiom,
    ! [X: nat,D2: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D2 ) @ ( vEBT_VEBT_low @ X @ D2 ) @ D2 )
      = X ) ).

% bit_split_inv
thf(fact_1163_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(6)
thf(fact_1164_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_1165_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_1166_power__add__numeral2,axiom,
    ! [A: complex,M: num,N: num,B: complex] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( 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 @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_1167_power__add__numeral2,axiom,
    ! [A: code_integer,M: num,N: num,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_1168_power__add__numeral2,axiom,
    ! [A: real,M: num,N: num,B: real] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( 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 @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_1169_power__add__numeral2,axiom,
    ! [A: rat,M: num,N: num,B: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( 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 @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_1170_power__add__numeral2,axiom,
    ! [A: nat,M: num,N: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( 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 @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_1171_power__add__numeral2,axiom,
    ! [A: int,M: num,N: num,B: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( 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 @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_1172_power__add__numeral,axiom,
    ! [A: complex,M: num,N: num] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_1173_power__add__numeral,axiom,
    ! [A: code_integer,M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_1174_power__add__numeral,axiom,
    ! [A: real,M: num,N: num] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_1175_power__add__numeral,axiom,
    ! [A: rat,M: num,N: num] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_1176_power__add__numeral,axiom,
    ! [A: nat,M: num,N: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_1177_power__add__numeral,axiom,
    ! [A: int,M: num,N: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_1178_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_1179_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_1180_add__One__commute,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ N )
      = ( plus_plus_num @ N @ one ) ) ).

% add_One_commute
thf(fact_1181_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ M @ N )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        & ( N = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
thf(fact_1182_zdiv__mult__self,axiom,
    ! [M: int,A: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ M @ N ) ) @ M )
        = ( plus_plus_int @ ( divide_divide_int @ A @ M ) @ N ) ) ) ).

% zdiv_mult_self
thf(fact_1183_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_1184_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_1185_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_1186_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_1187_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_1188_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_1189_zmult__zless__mono2,axiom,
    ! [I: int,J2: int,K: int] :
      ( ( ord_less_int @ I @ J2 )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J2 ) ) ) ) ).

% zmult_zless_mono2
thf(fact_1190_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_1191_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_1192_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_1193_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_1194_divides__aux__eq,axiom,
    ! [Q2: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_1195_divides__aux__eq,axiom,
    ! [Q2: code_integer,R2: code_integer] :
      ( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
      = ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
thf(fact_1196_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X2: nat,N4: nat] : ( modulo_modulo_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% low_def
thf(fact_1197_mlex__snd__decrI,axiom,
    ! [A: nat,A5: nat,B: nat,B4: nat,N5: nat] :
      ( ( A = A5 )
     => ( ( ord_less_nat @ B @ B4 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N5 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N5 ) @ B4 ) ) ) ) ).

% mlex_snd_decrI
thf(fact_1198_mlex__fst__decrI,axiom,
    ! [A: nat,A5: nat,B: nat,N5: nat,B4: nat] :
      ( ( ord_less_nat @ A @ A5 )
     => ( ( ord_less_nat @ B @ N5 )
       => ( ( ord_less_nat @ B4 @ N5 )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N5 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N5 ) @ B4 ) ) ) ) ) ).

% mlex_fst_decrI
thf(fact_1199_mlex__bound,axiom,
    ! [A: nat,A2: nat,B: nat,N5: nat] :
      ( ( ord_less_nat @ A @ A2 )
     => ( ( ord_less_nat @ B @ N5 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N5 ) @ B ) @ ( times_times_nat @ A2 @ N5 ) ) ) ) ).

% mlex_bound
thf(fact_1200_option_Osize__gen_I2_J,axiom,
    ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1201_option_Osize__gen_I2_J,axiom,
    ! [X: nat > nat,X22: nat] :
      ( ( size_option_nat @ X @ ( some_nat @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1202_option_Osize__gen_I2_J,axiom,
    ! [X: num > nat,X22: num] :
      ( ( size_option_num @ X @ ( some_num @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1203_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( suc @ zero_zero_nat ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( suc @ zero_zero_nat ) ) )
         => ~ ! [N2: nat] :
                ( ( X
                  = ( suc @ ( suc @ N2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
thf(fact_1204_uint32_Osize__eq,axiom,
    ( size_size_uint32
    = ( ^ [P3: uint32] : ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% uint32.size_eq
thf(fact_1205_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D2: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
         != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1206_add__scale__eq__noteq,axiom,
    ! [R2: code_integer,A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( R2 != zero_z3403309356797280102nteger )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ R2 @ C ) )
         != ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1207_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D2: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1208_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D2: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1209_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D2: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1210_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D2: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1211_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_1212_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_1213_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_1214_mult__less__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
        = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_1215_num_Osize__gen_I2_J,axiom,
    ! [X22: num] :
      ( ( size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_1216_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_1217_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_1218_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_1219_dvd__0__right,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).

% dvd_0_right
thf(fact_1220_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_1221_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_1222_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_1223_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_1224_dvd__0__right,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).

% dvd_0_right
thf(fact_1225_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
      = ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
thf(fact_1226_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_1227_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_1228_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_1229_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_1230_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
thf(fact_1231_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_1232_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_1233_mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_1234_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_1235_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_1236_mod__by__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% mod_by_0
thf(fact_1237_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_1238_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_1239_mod__self,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_1240_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_1241_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_1242_bits__mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_1243_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_1244_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_1245_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_1246_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_1247_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_1248_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_1249_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_1250_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_1251_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_1252_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_1253_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self1
thf(fact_1254_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_1255_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_1256_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self2
thf(fact_1257_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_1258_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_1259_nat__mod__eq_H,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ A @ N )
     => ( ( modulo_modulo_nat @ A @ N )
        = A ) ) ).

% nat_mod_eq'
thf(fact_1260_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = M ) ) ).

% mod_less
thf(fact_1261_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_1262_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_1263_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_1264_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_1265_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_1266_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_1267_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_1268_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_1269_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_1270_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_1271_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_1272_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_1273_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_1274_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_1275_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_1276_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_1277_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_1278_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_1279_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_1280_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_1281_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_1282_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_1283_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_1284_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_1285_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_1286_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_1287_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_1288_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_1289_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_1290_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_1291_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_1292_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_1293_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_1294_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_1295_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_1296_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_1297_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_1298_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_1299_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self4
thf(fact_1300_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_1301_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_1302_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self3
thf(fact_1303_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_1304_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_1305_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self2
thf(fact_1306_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_1307_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_1308_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self1
thf(fact_1309_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_1310_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_1311_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_1312_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_1313_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_1314_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_1315_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_1316_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_1317_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_1318_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_1319_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( M
        = ( suc @ zero_zero_nat ) ) ) ).

% dvd_1_iff_1
thf(fact_1320_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_1321_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_1322_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self1
thf(fact_1323_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self2
thf(fact_1324_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self3
thf(fact_1325_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self4
thf(fact_1326_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_1327_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_1328_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_1329_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_1330_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_1331_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_1332_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_1333_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_1334_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_1335_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_1336_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_1337_mod2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% mod2_Suc_Suc
thf(fact_1338_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_1339_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_1340_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_1341_add__self__mod__2,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% add_self_mod_2
thf(fact_1342_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_1343_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_1344_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_1345_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_1346_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_1347_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_1348_power__less__zero__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ zero_z3403309356797280102nteger )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ) ).

% power_less_zero_eq
thf(fact_1349_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_1350_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_1351_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_1352_power__less__zero__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) @ zero_z3403309356797280102nteger )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_1353_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_1354_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_1355_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_1356_zero__less__power__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ 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_z3403309356797280102nteger ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_1357_int__distrib_I1_J,axiom,
    ! [Z1: int,Z2: int,W: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).

% int_distrib(1)
thf(fact_1358_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z2: int] :
      ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
      = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).

% int_distrib(2)
thf(fact_1359_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_1360_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_1361_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_1362_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A4: nat,B2: nat] :
          ( ( modulo_modulo_nat @ B2 @ A4 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_1363_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A4: int,B2: int] :
          ( ( modulo_modulo_int @ B2 @ A4 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_1364_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A4: code_integer,B2: code_integer] :
          ( ( modulo364778990260209775nteger @ B2 @ A4 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_1365_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_1366_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_1367_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_1368_dvd__mod,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_1369_dvd__mod,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ K @ M )
     => ( ( dvd_dvd_int @ K @ N )
       => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_1370_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N: code_integer] :
      ( ( dvd_dvd_Code_integer @ K @ M )
     => ( ( dvd_dvd_Code_integer @ K @ N )
       => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_1371_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_1372_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_1373_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_1374_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_1375_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_1376_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_1377_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_1378_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_1379_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_1380_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_1381_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_1382_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_1383_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_1384_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ N )
     => ( ( dvd_dvd_nat @ N @ M )
       => ( M = N ) ) ) ).

% dvd_antisym
thf(fact_1385_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
      = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_1386_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_1387_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_1388_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_1389_mod__add__cong,axiom,
    ! [A: nat,C: nat,A5: nat,B: nat,B4: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A5 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B4 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1390_mod__add__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B4: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A5 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1391_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A5 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1392_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_1393_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_1394_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_1395_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_1396_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_1397_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_1398_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_1399_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_1400_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_1401_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A5: nat,B: nat,B4: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A5 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B4 @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1402_mod__mult__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B4: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A5 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1403_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A5 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1404_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_1405_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_1406_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_1407_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_1408_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_1409_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_1410_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_1411_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_1412_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_1413_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_1414_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_1415_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_1416_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_1417_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_1418_power__mod,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_1419_dvd__field__iff,axiom,
    ( dvd_dvd_complex
    = ( ^ [A4: complex,B2: complex] :
          ( ( A4 = zero_zero_complex )
         => ( B2 = zero_zero_complex ) ) ) ) ).

% dvd_field_iff
thf(fact_1420_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A4: real,B2: real] :
          ( ( A4 = zero_zero_real )
         => ( B2 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_1421_dvd__field__iff,axiom,
    ( dvd_dvd_rat
    = ( ^ [A4: rat,B2: rat] :
          ( ( A4 = zero_zero_rat )
         => ( B2 = zero_zero_rat ) ) ) ) ).

% dvd_field_iff
thf(fact_1422_dvd__0__left,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
thf(fact_1423_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_1424_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_1425_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_1426_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_1427_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
thf(fact_1428_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% mod_Suc_eq
thf(fact_1429_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).

% mod_Suc_Suc_eq
thf(fact_1430_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_1431_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_1432_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_1433_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_1434_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_1435_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_1436_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_1437_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_1438_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_1439_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_1440_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_1441_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_1442_dvdE,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ~ ! [K2: real] :
            ( A
           != ( times_times_real @ B @ K2 ) ) ) ).

% dvdE
thf(fact_1443_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ~ ! [K2: rat] :
            ( A
           != ( times_times_rat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_1444_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K2: nat] :
            ( A
           != ( times_times_nat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_1445_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K2: int] :
            ( A
           != ( times_times_int @ B @ K2 ) ) ) ).

% dvdE
thf(fact_1446_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A
        = ( times_times_real @ B @ K ) )
     => ( dvd_dvd_real @ B @ A ) ) ).

% dvdI
thf(fact_1447_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A
        = ( times_times_rat @ B @ K ) )
     => ( dvd_dvd_rat @ B @ A ) ) ).

% dvdI
thf(fact_1448_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_1449_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_1450_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B2: real,A4: real] :
        ? [K3: real] :
          ( A4
          = ( times_times_real @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_1451_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B2: rat,A4: rat] :
        ? [K3: rat] :
          ( A4
          = ( times_times_rat @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_1452_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B2: nat,A4: nat] :
        ? [K3: nat] :
          ( A4
          = ( times_times_nat @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_1453_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B2: int,A4: int] :
        ? [K3: int] :
          ( A4
          = ( times_times_int @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_1454_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_1455_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_1456_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_1457_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_1458_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_1459_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_1460_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_1461_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_1462_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_1463_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_1464_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_1465_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_1466_dvd__triv__left,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).

% dvd_triv_left
thf(fact_1467_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_1468_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_1469_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_1470_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ C @ D2 )
       => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_1471_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ C @ D2 )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_1472_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ C @ D2 )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_1473_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ C @ D2 )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_1474_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_1475_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_1476_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_1477_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_1478_dvd__triv__right,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).

% dvd_triv_right
thf(fact_1479_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_1480_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_1481_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_1482_nat__mod__eq,axiom,
    ! [B: nat,N: nat,A: nat] :
      ( ( ord_less_nat @ B @ N )
     => ( ( ( modulo_modulo_nat @ A @ N )
          = ( modulo_modulo_nat @ B @ N ) )
       => ( ( modulo_modulo_nat @ A @ N )
          = B ) ) ) ).

% nat_mod_eq
thf(fact_1483_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_1484_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_1485_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_1486_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_1487_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_1488_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_1489_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_1490_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_1491_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_1492_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_1493_div__div__div__same,axiom,
    ! [D2: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ D2 @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D2 ) @ ( divide_divide_nat @ B @ D2 ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_1494_div__div__div__same,axiom,
    ! [D2: int,B: int,A: int] :
      ( ( dvd_dvd_int @ D2 @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ ( divide_divide_int @ A @ D2 ) @ ( divide_divide_int @ B @ D2 ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_1495_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_1496_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_1497_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_1498_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_1499_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_1500_dvd__power__same,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( dvd_dvd_rat @ X @ Y )
     => ( dvd_dvd_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_1501_mod__plus__right,axiom,
    ! [A: nat,X: nat,M: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ X ) @ M )
        = ( modulo_modulo_nat @ ( plus_plus_nat @ B @ X ) @ M ) )
      = ( ( modulo_modulo_nat @ A @ M )
        = ( modulo_modulo_nat @ B @ M ) ) ) ).

% mod_plus_right
thf(fact_1502_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_1503_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_1504_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_1505_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_1506_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_1507_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1508_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1509_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1510_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1511_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1512_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1513_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1514_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_1515_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_1516_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = A )
      = ( ( divide6298287555418463151nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_1517_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_1518_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
     => ~ ! [D3: code_integer] :
            ( B
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_1519_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_1520_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_1521_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_1522_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_1523_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_1524_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_1525_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_1526_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_1527_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_1528_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
          = N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
         != N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).

% mod_Suc
thf(fact_1529_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P4: nat,M: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P4 )
       => ( ( ord_less_nat @ M @ P4 )
         => ( ! [N2: nat] :
                ( ( ord_less_nat @ N2 @ P4 )
               => ( ( P @ N2 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P4 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_1530_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_1531_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_1532_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_1533_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_1534_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_1535_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_1536_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_1537_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_1538_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_1539_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_1540_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_1541_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_1542_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_1543_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_1544_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D2: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( dvd_dvd_nat @ D2 @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D2 ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_1545_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D2: int,C: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( dvd_dvd_int @ D2 @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D2 ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_1546_nat__mod__lem,axiom,
    ! [N: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ B @ N )
        = ( ( modulo_modulo_nat @ B @ N )
          = B ) ) ) ).

% nat_mod_lem
thf(fact_1547_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_less_divisor
thf(fact_1548_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_1549_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_1550_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_1551_word__rot__lem,axiom,
    ! [L: nat,K: nat,D2: nat,N: nat] :
      ( ( ( plus_plus_nat @ L @ K )
        = ( plus_plus_nat @ D2 @ ( modulo_modulo_nat @ K @ L ) ) )
     => ( ( ord_less_nat @ N @ L )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ D2 @ N ) @ L )
          = N ) ) ) ).

% word_rot_lem
thf(fact_1552_mod__eq__0D,axiom,
    ! [M: nat,D2: nat] :
      ( ( ( modulo_modulo_nat @ M @ D2 )
        = zero_zero_nat )
     => ? [Q3: nat] :
          ( M
          = ( times_times_nat @ D2 @ Q3 ) ) ) ).

% mod_eq_0D
thf(fact_1553_msrevs_I2_J,axiom,
    ! [K: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) @ N )
      = ( modulo_modulo_nat @ M @ N ) ) ).

% msrevs(2)
thf(fact_1554_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_1555_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_1556_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1557_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1558_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1559_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_1560_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_1561_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_1562_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1563_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1564_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1565_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1566_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1567_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1568_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_1569_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_1570_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_1571_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_1572_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_1573_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_1574_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_1575_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_1576_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] :
      ( A
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_1577_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_1578_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_1579_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_1580_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_1581_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_1582_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_1583_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_1584_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_1585_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_1586_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_1587_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_1588_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_1589_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_1590_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_1591_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_1592_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_1593_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_1594_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_1595_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_1596_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_1597_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_1598_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_1599_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_1600_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_1601_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_1602_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_1603_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( ( dvd_dvd_Code_integer @ A @ B )
         => ( ( dvd_dvd_Code_integer @ C @ D2 )
           => ( ( ( divide6298287555418463151nteger @ B @ A )
                = ( divide6298287555418463151nteger @ D2 @ C ) )
              = ( ( times_3573771949741848930nteger @ B @ C )
                = ( times_3573771949741848930nteger @ A @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_1604_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B )
         => ( ( dvd_dvd_nat @ C @ D2 )
           => ( ( ( divide_divide_nat @ B @ A )
                = ( divide_divide_nat @ D2 @ C ) )
              = ( ( times_times_nat @ B @ C )
                = ( times_times_nat @ A @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_1605_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B )
         => ( ( dvd_dvd_int @ C @ D2 )
           => ( ( ( divide_divide_int @ B @ A )
                = ( divide_divide_int @ D2 @ C ) )
              = ( ( times_times_int @ B @ C )
                = ( times_times_int @ A @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_1606_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).

% mod_mult2_eq
thf(fact_1607_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_mult_cancel
thf(fact_1608_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_1609_bex2I,axiom,
    ! [A: heap_e7401611519738050253t_unit,B: set_nat,S2: set_Pr3948176798113811640et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
      ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A @ B ) @ S2 )
     => ( ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A6: heap_e7401611519738050253t_unit,B5: set_nat] :
            ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A6 @ B5 ) @ S2 )
            & ( P @ A6 @ B5 ) ) ) ) ).

% bex2I
thf(fact_1610_bex2I,axiom,
    ! [A: num,B: num,S2: set_Pr8218934625190621173um_num,P: num > num > $o] :
      ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A @ B ) @ S2 )
     => ( ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A6: num,B5: num] :
            ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A6 @ B5 ) @ S2 )
            & ( P @ A6 @ B5 ) ) ) ) ).

% bex2I
thf(fact_1611_bex2I,axiom,
    ! [A: nat,B: num,S2: set_Pr6200539531224447659at_num,P: nat > num > $o] :
      ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ S2 )
     => ( ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A6: nat,B5: num] :
            ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A6 @ B5 ) @ S2 )
            & ( P @ A6 @ B5 ) ) ) ) ).

% bex2I
thf(fact_1612_bex2I,axiom,
    ! [A: nat,B: nat,S2: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S2 )
     => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A6: nat,B5: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B5 ) @ S2 )
            & ( P @ A6 @ B5 ) ) ) ) ).

% bex2I
thf(fact_1613_bex2I,axiom,
    ! [A: int,B: int,S2: set_Pr958786334691620121nt_int,P: int > int > $o] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ S2 )
     => ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A6: int,B5: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A6 @ B5 ) @ S2 )
            & ( P @ A6 @ B5 ) ) ) ) ).

% bex2I
thf(fact_1614_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_1615_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_1616_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_1617_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_1618_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_1619_evenE,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: nat] :
            ( A
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_1620_evenE,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: int] :
            ( A
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_1621_bit__eq__rec,axiom,
    ( ( ^ [Y6: nat,Z3: nat] : Y6 = Z3 )
    = ( ^ [A4: nat,B2: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_1622_bit__eq__rec,axiom,
    ( ( ^ [Y6: int,Z3: int] : Y6 = Z3 )
    = ( ^ [A4: int,B2: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_1623_mod__lemma,axiom,
    ! [C: nat,R2: nat,B: nat,Q2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_nat @ R2 @ B )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ Q2 @ C ) ) @ R2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mod_lemma
thf(fact_1624_split__mod,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_1625_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_1626_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_1627_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_1628_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_1629_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_1630_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_1631_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_1632_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_1633_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_1634_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
          = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_1635_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_1636_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_1637_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_1638_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: nat,M: 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 ) ) @ M ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1639_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M: 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 ) ) @ M ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1640_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1641_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_1642_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_1643_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_1644_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B
        = ( plus_plus_complex @ B @ A ) )
      = ( A = zero_zero_complex ) ) ).

% add_0_iff
thf(fact_1645_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_1646_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_1647_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_1648_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_1649_add__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B
        = ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% add_0_iff
thf(fact_1650_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_1651_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_1652_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_1653_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_1654_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ( A != B )
        & ( C != D2 ) )
      = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D2 ) )
       != ( plus_plus_real @ ( times_times_real @ A @ D2 ) @ ( times_times_real @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_1655_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( A != B )
        & ( C != D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D2 ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_1656_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ( A != B )
        & ( C != D2 ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_1657_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( A != B )
        & ( C != D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_1658_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_1659_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_1660_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_1661_zero__less__power__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_z3403309356797280102nteger ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_1662_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_1663_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_1664_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_1665_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N ) ) )
          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N ) ) )
          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
thf(fact_1666_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_1667_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_1668_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_1669_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_1670_div__mod__decomp,axiom,
    ! [A2: nat,N: nat] :
      ( A2
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).

% div_mod_decomp
thf(fact_1671_div__less__mono,axiom,
    ! [A2: nat,B3: nat,N: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A2 @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B3 @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B3 @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_1672_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X3: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D3 @ A )
          & ( dvd_dvd_nat @ D3 @ B )
          & ( ( times_times_nat @ A @ X3 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_1673_unity__coeff__ex,axiom,
    ! [P: complex > $o,L: complex] :
      ( ( ? [X2: complex] : ( P @ ( times_times_complex @ L @ X2 ) ) )
      = ( ? [X2: complex] :
            ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X2 @ zero_zero_complex ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_1674_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L: code_integer] :
      ( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X2 ) ) )
      = ( ? [X2: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_1675_unity__coeff__ex,axiom,
    ! [P: real > $o,L: real] :
      ( ( ? [X2: real] : ( P @ ( times_times_real @ L @ X2 ) ) )
      = ( ? [X2: real] :
            ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X2 @ zero_zero_real ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_1676_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X2: rat] : ( P @ ( times_times_rat @ L @ X2 ) ) )
      = ( ? [X2: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_1677_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X2: nat] : ( P @ ( times_times_nat @ L @ X2 ) ) )
      = ( ? [X2: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_1678_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X2: int] : ( P @ ( times_times_int @ L @ X2 ) ) )
      = ( ? [X2: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X2 @ zero_zero_int ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_1679_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_1680_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_1681_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_1682_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_1683_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_1684_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_1685_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y2: num] :
      ( ( ( bit0 @ X22 )
        = ( bit0 @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% verit_eq_simplify(8)
thf(fact_1686_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_1687_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_1688_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_1689_mod__plus__cong,axiom,
    ! [B: int,B4: int,X: int,X6: int,Y: int,Y7: int,Z4: int] :
      ( ( B = B4 )
     => ( ( ( modulo_modulo_int @ X @ B4 )
          = ( modulo_modulo_int @ X6 @ B4 ) )
       => ( ( ( modulo_modulo_int @ Y @ B4 )
            = ( modulo_modulo_int @ Y7 @ B4 ) )
         => ( ( ( plus_plus_int @ X6 @ Y7 )
              = Z4 )
           => ( ( modulo_modulo_int @ ( plus_plus_int @ X @ Y ) @ B )
              = ( modulo_modulo_int @ Z4 @ B4 ) ) ) ) ) ) ).

% mod_plus_cong
thf(fact_1690_zmod__helper,axiom,
    ! [N: int,M: int,K: int,A: int] :
      ( ( ( modulo_modulo_int @ N @ M )
        = K )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ N @ A ) @ M )
        = ( modulo_modulo_int @ ( plus_plus_int @ K @ A ) @ M ) ) ) ).

% zmod_helper
thf(fact_1691_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( ( dvd_dvd_int @ M @ T )
        = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).

% zdvd_mono
thf(fact_1692_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_1693_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).

% neg_mod_bound
thf(fact_1694_zmod__eq__0D,axiom,
    ! [M: int,D2: int] :
      ( ( ( modulo_modulo_int @ M @ D2 )
        = zero_zero_int )
     => ? [Q3: int] :
          ( M
          = ( times_times_int @ D2 @ Q3 ) ) ) ).

% zmod_eq_0D
thf(fact_1695_zmod__eq__0__iff,axiom,
    ! [M: int,D2: int] :
      ( ( ( modulo_modulo_int @ M @ D2 )
        = zero_zero_int )
      = ( ? [Q4: int] :
            ( M
            = ( times_times_int @ D2 @ Q4 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_1696_zdiv__mono__strict,axiom,
    ! [A2: int,B3: int,N: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ( ( modulo_modulo_int @ A2 @ N )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B3 @ N )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B3 @ N ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_1697_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_int @ M @ N )
       => ~ ( dvd_dvd_int @ N @ M ) ) ) ).

% zdvd_not_zless
thf(fact_1698_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
     => ( ( K != zero_zero_int )
       => ( dvd_dvd_int @ M @ N ) ) ) ).

% zdvd_mult_cancel
thf(fact_1699_div__mod__decomp__int,axiom,
    ! [A2: int,N: int] :
      ( A2
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).

% div_mod_decomp_int
thf(fact_1700_zdvd__period,axiom,
    ! [A: int,D2: int,X: int,T: int,C: int] :
      ( ( dvd_dvd_int @ A @ D2 )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D2 ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_1701_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
      = ( dvd_dvd_int @ K @ N ) ) ).

% zdvd_reduce
thf(fact_1702_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1703_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1704_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1705_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1706_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1707_verit__comp__simplify1_I1_J,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1708_pinf_I1_J,axiom,
    ! [P: real > $o,P2: real > $o,Q: real > $o,Q5: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z6 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1709_pinf_I1_J,axiom,
    ! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q5: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z6 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1710_pinf_I1_J,axiom,
    ! [P: num > $o,P2: num > $o,Q: num > $o,Q5: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z6 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1711_pinf_I1_J,axiom,
    ! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q5: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z6 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1712_pinf_I1_J,axiom,
    ! [P: int > $o,P2: int > $o,Q: int > $o,Q5: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z6 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1713_pinf_I1_J,axiom,
    ! [P: code_integer > $o,P2: code_integer > $o,Q: code_integer > $o,Q5: code_integer > $o] :
      ( ? [Z5: code_integer] :
        ! [X3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: code_integer] :
          ! [X3: code_integer] :
            ( ( ord_le6747313008572928689nteger @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: code_integer] :
          ! [X4: code_integer] :
            ( ( ord_le6747313008572928689nteger @ Z6 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1714_pinf_I2_J,axiom,
    ! [P: real > $o,P2: real > $o,Q: real > $o,Q5: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z6 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1715_pinf_I2_J,axiom,
    ! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q5: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z6 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1716_pinf_I2_J,axiom,
    ! [P: num > $o,P2: num > $o,Q: num > $o,Q5: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z6 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1717_pinf_I2_J,axiom,
    ! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q5: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z6 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1718_pinf_I2_J,axiom,
    ! [P: int > $o,P2: int > $o,Q: int > $o,Q5: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z6 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1719_pinf_I2_J,axiom,
    ! [P: code_integer > $o,P2: code_integer > $o,Q: code_integer > $o,Q5: code_integer > $o] :
      ( ? [Z5: code_integer] :
        ! [X3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: code_integer] :
          ! [X3: code_integer] :
            ( ( ord_le6747313008572928689nteger @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: code_integer] :
          ! [X4: code_integer] :
            ( ( ord_le6747313008572928689nteger @ Z6 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1720_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z6 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1721_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z6 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1722_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z6: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z6 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1723_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z6 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1724_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z6 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1725_pinf_I3_J,axiom,
    ! [T: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z6 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_1726_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z6 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1727_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z6 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1728_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z6: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z6 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1729_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z6 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1730_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z6 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1731_pinf_I4_J,axiom,
    ! [T: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z6 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_1732_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z6 @ X4 )
     => ~ ( ord_less_real @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1733_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z6 @ X4 )
     => ~ ( ord_less_rat @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1734_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z6: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z6 @ X4 )
     => ~ ( ord_less_num @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1735_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z6 @ X4 )
     => ~ ( ord_less_nat @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1736_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z6 @ X4 )
     => ~ ( ord_less_int @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1737_pinf_I5_J,axiom,
    ! [T: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z6 @ X4 )
     => ~ ( ord_le6747313008572928689nteger @ X4 @ T ) ) ).

% pinf(5)
thf(fact_1738_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z6 @ X4 )
     => ( ord_less_real @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1739_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z6 @ X4 )
     => ( ord_less_rat @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1740_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z6: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z6 @ X4 )
     => ( ord_less_num @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1741_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z6 @ X4 )
     => ( ord_less_nat @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1742_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z6 @ X4 )
     => ( ord_less_int @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1743_pinf_I7_J,axiom,
    ! [T: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z6 @ X4 )
     => ( ord_le6747313008572928689nteger @ T @ X4 ) ) ).

% pinf(7)
thf(fact_1744_minf_I1_J,axiom,
    ! [P: real > $o,P2: real > $o,Q: real > $o,Q5: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z6 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1745_minf_I1_J,axiom,
    ! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q5: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z6 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1746_minf_I1_J,axiom,
    ! [P: num > $o,P2: num > $o,Q: num > $o,Q5: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z6 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1747_minf_I1_J,axiom,
    ! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q5: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z6 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1748_minf_I1_J,axiom,
    ! [P: int > $o,P2: int > $o,Q: int > $o,Q5: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z6 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1749_minf_I1_J,axiom,
    ! [P: code_integer > $o,P2: code_integer > $o,Q: code_integer > $o,Q5: code_integer > $o] :
      ( ? [Z5: code_integer] :
        ! [X3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: code_integer] :
          ! [X3: code_integer] :
            ( ( ord_le6747313008572928689nteger @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: code_integer] :
          ! [X4: code_integer] :
            ( ( ord_le6747313008572928689nteger @ X4 @ Z6 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                & ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1750_minf_I2_J,axiom,
    ! [P: real > $o,P2: real > $o,Q: real > $o,Q5: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z6 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1751_minf_I2_J,axiom,
    ! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q5: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z6 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1752_minf_I2_J,axiom,
    ! [P: num > $o,P2: num > $o,Q: num > $o,Q5: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z6 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1753_minf_I2_J,axiom,
    ! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q5: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z6 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1754_minf_I2_J,axiom,
    ! [P: int > $o,P2: int > $o,Q: int > $o,Q5: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z6 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1755_minf_I2_J,axiom,
    ! [P: code_integer > $o,P2: code_integer > $o,Q: code_integer > $o,Q5: code_integer > $o] :
      ( ? [Z5: code_integer] :
        ! [X3: code_integer] :
          ( ( ord_le6747313008572928689nteger @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P2 @ X3 ) ) )
     => ( ? [Z5: code_integer] :
          ! [X3: code_integer] :
            ( ( ord_le6747313008572928689nteger @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q5 @ X3 ) ) )
       => ? [Z6: code_integer] :
          ! [X4: code_integer] :
            ( ( ord_le6747313008572928689nteger @ X4 @ Z6 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P2 @ X4 )
                | ( Q5 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1756_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z6 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1757_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z6 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1758_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z6: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z6 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1759_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z6 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1760_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z6 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1761_minf_I3_J,axiom,
    ! [T: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z6 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_1762_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z6 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1763_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z6 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1764_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z6: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z6 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1765_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z6 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1766_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z6 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1767_minf_I4_J,axiom,
    ! [T: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z6 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_1768_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z6 )
     => ( ord_less_real @ X4 @ T ) ) ).

% minf(5)
thf(fact_1769_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z6 )
     => ( ord_less_rat @ X4 @ T ) ) ).

% minf(5)
thf(fact_1770_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z6: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z6 )
     => ( ord_less_num @ X4 @ T ) ) ).

% minf(5)
thf(fact_1771_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z6 )
     => ( ord_less_nat @ X4 @ T ) ) ).

% minf(5)
thf(fact_1772_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z6 )
     => ( ord_less_int @ X4 @ T ) ) ).

% minf(5)
thf(fact_1773_minf_I5_J,axiom,
    ! [T: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z6 )
     => ( ord_le6747313008572928689nteger @ X4 @ T ) ) ).

% minf(5)
thf(fact_1774_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z6 )
     => ~ ( ord_less_real @ T @ X4 ) ) ).

% minf(7)
thf(fact_1775_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z6 )
     => ~ ( ord_less_rat @ T @ X4 ) ) ).

% minf(7)
thf(fact_1776_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z6: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z6 )
     => ~ ( ord_less_num @ T @ X4 ) ) ).

% minf(7)
thf(fact_1777_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z6 )
     => ~ ( ord_less_nat @ T @ X4 ) ) ).

% minf(7)
thf(fact_1778_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z6 )
     => ~ ( ord_less_int @ T @ X4 ) ) ).

% minf(7)
thf(fact_1779_minf_I7_J,axiom,
    ! [T: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z6 )
     => ~ ( ord_le6747313008572928689nteger @ T @ X4 ) ) ).

% minf(7)
thf(fact_1780_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M3: nat] :
          ( X
         != ( product_Pair_nat_nat @ K2 @ M3 ) ) ).

% prod_decode_aux.cases
thf(fact_1781_mod__exp__less__eq__exp,axiom,
    ! [A: int,N: nat] : ( ord_less_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% mod_exp_less_eq_exp
thf(fact_1782_pos__mod__bound2,axiom,
    ! [A: int] : ( ord_less_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% pos_mod_bound2
thf(fact_1783_verit__sum__simplify,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% verit_sum_simplify
thf(fact_1784_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_1785_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_1786_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_1787_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_1788_verit__sum__simplify,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% verit_sum_simplify
thf(fact_1789_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] :
      ( one
     != ( bit0 @ X22 ) ) ).

% verit_eq_simplify(10)
thf(fact_1790_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va: nat] :
              ( X
             != ( suc @ ( suc @ Va ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_1791_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N2: nat] :
            ( X
           != ( suc @ N2 ) ) ) ).

% list_decode.cases
thf(fact_1792_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
     => ? [B6: nat,C3: nat] :
          ( ( A
            = ( times_times_nat @ B6 @ C3 ) )
          & ( dvd_dvd_nat @ B6 @ B )
          & ( dvd_dvd_nat @ C3 @ C ) ) ) ).

% division_decomp
thf(fact_1793_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
     => ? [B6: int,C3: int] :
          ( ( A
            = ( times_times_int @ B6 @ C3 ) )
          & ( dvd_dvd_int @ B6 @ B )
          & ( dvd_dvd_int @ C3 @ C ) ) ) ).

% division_decomp
thf(fact_1794_dvd__productE,axiom,
    ! [P4: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A @ B ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P4
              = ( times_times_nat @ X3 @ Y3 ) )
           => ( ( dvd_dvd_nat @ X3 @ A )
             => ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_1795_dvd__productE,axiom,
    ! [P4: int,A: int,B: int] :
      ( ( dvd_dvd_int @ P4 @ ( times_times_int @ A @ B ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( P4
              = ( times_times_int @ X3 @ Y3 ) )
           => ( ( dvd_dvd_int @ X3 @ A )
             => ~ ( dvd_dvd_int @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_1796_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A6: nat,B5: nat] :
          ( ( P @ A6 @ B5 )
          = ( P @ B5 @ A6 ) )
     => ( ! [A6: nat] : ( P @ A6 @ zero_zero_nat )
       => ( ! [A6: nat,B5: nat] :
              ( ( P @ A6 @ B5 )
             => ( P @ A6 @ ( plus_plus_nat @ A6 @ B5 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_1797_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_1798_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
thf(fact_1799_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_1800_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
thf(fact_1801_gcd__nat_Oextremum,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_1802_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_1803_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_1804_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_1805_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_1806_pinf_I9_J,axiom,
    ! [D2: real,S: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z6 @ X4 )
     => ( ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X4 @ S ) )
        = ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_1807_pinf_I9_J,axiom,
    ! [D2: rat,S: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z6 @ X4 )
     => ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ S ) )
        = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_1808_pinf_I9_J,axiom,
    ! [D2: nat,S: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z6 @ X4 )
     => ( ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X4 @ S ) )
        = ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_1809_pinf_I9_J,axiom,
    ! [D2: int,S: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z6 @ X4 )
     => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ S ) )
        = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_1810_pinf_I9_J,axiom,
    ! [D2: code_integer,S: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z6 @ X4 )
     => ( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X4 @ S ) )
        = ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_1811_pinf_I10_J,axiom,
    ! [D2: real,S: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z6 @ X4 )
     => ( ( ~ ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_1812_pinf_I10_J,axiom,
    ! [D2: rat,S: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z6 @ X4 )
     => ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_1813_pinf_I10_J,axiom,
    ! [D2: nat,S: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z6 @ X4 )
     => ( ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_1814_pinf_I10_J,axiom,
    ! [D2: int,S: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z6 @ X4 )
     => ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_1815_pinf_I10_J,axiom,
    ! [D2: code_integer,S: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z6 @ X4 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_1816_minf_I9_J,axiom,
    ! [D2: real,S: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z6 )
     => ( ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X4 @ S ) )
        = ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_1817_minf_I9_J,axiom,
    ! [D2: rat,S: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z6 )
     => ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ S ) )
        = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_1818_minf_I9_J,axiom,
    ! [D2: nat,S: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z6 )
     => ( ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X4 @ S ) )
        = ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_1819_minf_I9_J,axiom,
    ! [D2: int,S: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z6 )
     => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ S ) )
        = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_1820_minf_I9_J,axiom,
    ! [D2: code_integer,S: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z6 )
     => ( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X4 @ S ) )
        = ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_1821_minf_I10_J,axiom,
    ! [D2: real,S: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z6 )
     => ( ( ~ ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_1822_minf_I10_J,axiom,
    ! [D2: rat,S: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z6 )
     => ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_1823_minf_I10_J,axiom,
    ! [D2: nat,S: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z6 )
     => ( ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_1824_minf_I10_J,axiom,
    ! [D2: int,S: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z6 )
     => ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_1825_minf_I10_J,axiom,
    ! [D2: code_integer,S: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z6 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_1826_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ M @ N )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_1827_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [M3: nat] : ( P @ M3 @ zero_zero_nat )
     => ( ! [M3: nat,N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ( P @ N2 @ ( modulo_modulo_nat @ M3 @ N2 ) )
             => ( P @ M3 @ N2 ) ) )
       => ( P @ M @ N ) ) ) ).

% gcd_nat_induct
thf(fact_1828_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) )
        | ( ( times_times_nat @ B @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D3 ) ) ) ) ).

% bezout_add_nat
thf(fact_1829_bezout__lemma__nat,axiom,
    ! [D2: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D2 @ A )
     => ( ( dvd_dvd_nat @ D2 @ B )
       => ( ( ( ( times_times_nat @ A @ X )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) )
            | ( ( times_times_nat @ B @ X )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D2 ) ) )
         => ? [X3: nat,Y3: nat] :
              ( ( dvd_dvd_nat @ D2 @ A )
              & ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D2 ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D2 ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_1830_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N4: nat] : ( divide_divide_nat @ ( times_times_nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_1831_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_1832_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_1833_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_1834_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_1835_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_1836_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_1837_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_1838_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_1839_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_1840_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_1841_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_1842_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_1843_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_1844_power__le__zero__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) @ zero_z3403309356797280102nteger )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_1845_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_1846_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_1847_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_1848_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_1849_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_1850_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numera6620942414471956472nteger @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_1851_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_1852_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_1853_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_rat @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_1854_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_1855_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_1856_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs2: set_nat,X2: nat] :
          ( ( member_nat @ X2 @ Xs2 )
          & ! [Y5: nat] :
              ( ( member_nat @ Y5 @ Xs2 )
             => ( ord_less_eq_nat @ X2 @ Y5 ) ) ) ) ) ).

% min_in_set_def
thf(fact_1857_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs2: set_nat,X2: nat] :
          ( ( member_nat @ X2 @ Xs2 )
          & ! [Y5: nat] :
              ( ( member_nat @ Y5 @ Xs2 )
             => ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).

% max_in_set_def
thf(fact_1858_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(71)
thf(fact_1859_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_1860_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_1861_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1862_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1863_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1864_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1865_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1866_power__one__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1867_power__one__right,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1868_power__one__right,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1869_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_1870_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N ) )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_1871_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_1872_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ one_one_nat )
      = ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
thf(fact_1873_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1874_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_1875_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_1876_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_1877_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_1878_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_1879_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_1880_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_1881_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_1882_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_1883_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_1884_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_1885_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_1886_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_1887_mult_Oright__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.right_neutral
thf(fact_1888_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_1889_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_1890_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_1891_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_1892_mult__1,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% mult_1
thf(fact_1893_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_1894_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_1895_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_1896_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_1897_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_1898_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_1899_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_1900_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_1901_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_1902_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_1903_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_1904_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_assn @ one_one_assn @ N )
      = one_one_assn ) ).

% power_one
thf(fact_1905_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_1906_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_real @ one_one_real @ N )
      = one_one_real ) ).

% power_one
thf(fact_1907_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_1908_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ one_one_complex @ N )
      = one_one_complex ) ).

% power_one
thf(fact_1909_power__one,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ one_one_Code_integer @ N )
      = one_one_Code_integer ) ).

% power_one
thf(fact_1910_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ one_one_rat @ N )
      = one_one_rat ) ).

% power_one
thf(fact_1911_less__eq__option__Some,axiom,
    ! [X: real,Y: real] :
      ( ( ord_le8614940839814719452n_real @ ( some_real @ X ) @ ( some_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1912_less__eq__option__Some,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_le2843612097646854710et_nat @ ( some_set_nat @ X ) @ ( some_set_nat @ Y ) )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1913_less__eq__option__Some,axiom,
    ! [X: num,Y: num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ X ) @ ( some_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1914_less__eq__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1915_less__eq__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ X ) @ ( some_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1916_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1917_signed__take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_ri6519982836138164636nteger @ N @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% signed_take_bit_of_0
thf(fact_1918_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_1919_of__bool__eq_I1_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $false )
    = zero_zero_complex ) ).

% of_bool_eq(1)
thf(fact_1920_of__bool__eq_I1_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $false )
    = zero_zero_real ) ).

% of_bool_eq(1)
thf(fact_1921_of__bool__eq_I1_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $false )
    = zero_zero_rat ) ).

% of_bool_eq(1)
thf(fact_1922_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_1923_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_1924_of__bool__eq_I1_J,axiom,
    ( ( zero_n356916108424825756nteger @ $false )
    = zero_z3403309356797280102nteger ) ).

% of_bool_eq(1)
thf(fact_1925_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = zero_zero_complex )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1926_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = zero_zero_real )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1927_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = zero_zero_rat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1928_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = zero_zero_nat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1929_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = zero_zero_int )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1930_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = zero_z3403309356797280102nteger )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1931_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_1932_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_1933_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_1934_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_1935_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_1936_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_1937_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_1938_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_1939_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_1940_of__bool__eq_I2_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $true )
    = one_one_real ) ).

% of_bool_eq(2)
thf(fact_1941_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_1942_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_1943_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_1944_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_1945_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = one_one_real )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_1946_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = one_one_rat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_1947_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_1948_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_1949_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = one_one_Code_integer )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_1950_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_1951_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_1952_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_1953_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_1954_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(1)
thf(fact_1955_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_1956_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y5: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X2 ) @ ( some_nat @ Y5 ) ) ) ) ).

% lesseq_shift
thf(fact_1957_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_1958_add__le__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_le_same_cancel1
thf(fact_1959_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_1960_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_1961_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_1962_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_1963_add__le__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_le_same_cancel2
thf(fact_1964_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_1965_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_1966_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_1967_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_1968_le__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% le_add_same_cancel1
thf(fact_1969_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_1970_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_1971_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_1972_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_1973_le__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% le_add_same_cancel2
thf(fact_1974_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_1975_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_1976_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_1977_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_1978_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1979_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_1980_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_1981_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_1982_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ A ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1983_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_1984_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_1985_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_1986_mult__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C
        = ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
thf(fact_1987_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_1988_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_1989_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_1990_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_1991_mult__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ( times_3573771949741848930nteger @ C @ A )
        = C )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
thf(fact_1992_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_1993_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_1994_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_1995_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_1996_mult__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C
        = ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
thf(fact_1997_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_1998_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_1999_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_2000_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_2001_mult__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ C )
        = C )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
thf(fact_2002_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_2003_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_2004_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_2005_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6690914467698888265omplex @ N )
        = one_one_complex )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_2006_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_real @ N )
        = one_one_real )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_2007_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_rat @ N )
        = one_one_rat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_2008_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_2009_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_2010_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_2011_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_2012_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_2013_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_2014_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_2015_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_2016_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_2017_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_2018_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ A @ A )
        = one_one_Code_integer ) ) ).

% div_self
thf(fact_2019_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_2020_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_2021_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_2022_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_2023_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_2024_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_2025_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_2026_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_2027_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_2028_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_2029_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_2030_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_2031_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_2032_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_2033_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_2034_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_2035_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_2036_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_2037_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_2038_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_2039_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_2040_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_2041_power__inject__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M )
          = ( power_power_real @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_2042_power__inject__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M )
          = ( power_power_rat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_2043_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_2044_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_2045_power__inject__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ( power_8256067586552552935nteger @ A @ M )
          = ( power_8256067586552552935nteger @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_2046_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_2047_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_2048_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_2049_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_2050_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_2051_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_2052_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_2053_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_2054_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_2055_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_2056_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_2057_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_2058_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_2059_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_2060_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_2061_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N )
          = M )
        = ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_2062_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% mult_le_cancel2
thf(fact_2063_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_2064_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_2065_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_2066_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_2067_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_2068_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_2069_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_2070_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_2071_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_2072_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_2073_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_2074_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_2075_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_numeral_of_1
thf(fact_2076_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_2077_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_2078_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_2079_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_2080_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_2081_int__div__same__is__1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ( divide_divide_int @ A @ B )
          = A )
        = ( B = one_one_int ) ) ) ).

% int_div_same_is_1
thf(fact_2082_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_2083_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_2084_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_2085_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_2086_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_2087_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_2088_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_2089_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_2090_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_2091_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_2092_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_2093_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_2094_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_2095_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_2096_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_2097_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_2098_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_2099_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_2100_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_2101_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_2102_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_2103_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_2104_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_2105_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_2106_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_2107_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_2108_power__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2109_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_2110_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_2111_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_2112_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_2113_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_2114_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_2115_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_2116_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_2117_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_2118_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_2119_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_2120_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_2121_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_2122_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_2123_power__strict__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2124_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_2125_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_2126_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_2127_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_2128_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_2129_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_2130_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_2131_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_2132_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_2133_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_2134_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_2135_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_2136_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_2137_power__decreasing__iff,axiom,
    ! [B: rat,M: 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 @ M ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2138_power__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ M ) @ ( power_8256067586552552935nteger @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2139_power__decreasing__iff,axiom,
    ! [B: real,M: 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 @ M ) @ ( power_power_real @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2140_power__decreasing__iff,axiom,
    ! [B: nat,M: 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 @ M ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2141_power__decreasing__iff,axiom,
    ! [B: int,M: 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 @ M ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2142_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2143_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2144_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2145_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2146_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2147_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2148_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2149_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2150_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2151_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ M ) @ ( power_8256067586552552935nteger @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2152_power__mono__iff,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) )
            = ( ord_le3102999989581377725nteger @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2153_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_2154_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_2155_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_2156_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_2157_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_2158_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_2159_one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_mod_two_eq_one
thf(fact_2160_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_2161_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_2162_bits__one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_one_mod_two_eq_one
thf(fact_2163_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_2164_odd__of__bool__self,axiom,
    ! [P4: $o] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P4 ) ) )
      = P4 ) ).

% odd_of_bool_self
thf(fact_2165_odd__of__bool__self,axiom,
    ! [P4: $o] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P4 ) ) )
      = P4 ) ).

% odd_of_bool_self
thf(fact_2166_odd__of__bool__self,axiom,
    ! [P4: $o] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P4 ) ) )
      = P4 ) ).

% odd_of_bool_self
thf(fact_2167_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_2168_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_2169_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_2170_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_2171_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_2172_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_2173_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_2174_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_2175_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_2176_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_2177_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% half_nonnegative_int_iff
thf(fact_2178_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_2179_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_2180_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_2181_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_2182_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_2183_one__div__two__eq__zero,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% one_div_two_eq_zero
thf(fact_2184_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_2185_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_2186_bits__1__div__2,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% bits_1_div_2
thf(fact_2187_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_2188_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_2189_power2__less__eq__zero__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
thf(fact_2190_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_2191_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_2192_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_2193_power2__eq__iff__nonneg,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2194_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_2195_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_2196_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_2197_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_2198_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_2199_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_2200_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_2201_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_2202_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_2203_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_2204_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_2205_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_2206_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_2207_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_2208_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = zero_z3403309356797280102nteger ) ).

% of_bool_half_eq_0
thf(fact_2209_mod2__gr__0,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% mod2_gr_0
thf(fact_2210_one__mod__exp__eq__one,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = one_one_int ) ).

% one_mod_exp_eq_one
thf(fact_2211_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_2212_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_2213_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_2214_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_2215_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_2216_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_2217_zero__le__power__eq__numeral,axiom,
    ! [A: code_integer,W: num] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ 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_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_2218_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_2219_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_2220_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_2221_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_2222_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_2223_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_2224_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_2225_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_2226_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_2227_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_2228_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_2229_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_2230_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_2231_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_2232_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).

% set_bit_greater_eq
thf(fact_2233_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_2234_not__one__le__zero,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_le_zero
thf(fact_2235_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_2236_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_2237_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_2238_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_2239_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2240_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_2241_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_2242_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_2243_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_2244_zero__less__one__class_Ozero__le__one,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one_class.zero_le_one
thf(fact_2245_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_2246_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_2247_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_2248_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_2249_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).

% one_le_numeral
thf(fact_2250_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_2251_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_2252_lift__Suc__mono__le,axiom,
    ! [F: nat > real,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2253_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2254_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2255_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2256_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2257_lift__Suc__antimono__le,axiom,
    ! [F: nat > real,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2258_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2259_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2260_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2261_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2262_one__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% one_le_power
thf(fact_2263_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_2264_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_2265_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_2266_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_2267_le__some__optE,axiom,
    ! [M: real,X: option_real] :
      ( ( ord_le8614940839814719452n_real @ ( some_real @ M ) @ X )
     => ~ ! [M6: real] :
            ( ( X
              = ( some_real @ M6 ) )
           => ~ ( ord_less_eq_real @ M @ M6 ) ) ) ).

% le_some_optE
thf(fact_2268_le__some__optE,axiom,
    ! [M: set_nat,X: option_set_nat] :
      ( ( ord_le2843612097646854710et_nat @ ( some_set_nat @ M ) @ X )
     => ~ ! [M6: set_nat] :
            ( ( X
              = ( some_set_nat @ M6 ) )
           => ~ ( ord_less_eq_set_nat @ M @ M6 ) ) ) ).

% le_some_optE
thf(fact_2269_le__some__optE,axiom,
    ! [M: num,X: option_num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ M ) @ X )
     => ~ ! [M6: num] :
            ( ( X
              = ( some_num @ M6 ) )
           => ~ ( ord_less_eq_num @ M @ M6 ) ) ) ).

% le_some_optE
thf(fact_2270_le__some__optE,axiom,
    ! [M: nat,X: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ M ) @ X )
     => ~ ! [M6: nat] :
            ( ( X
              = ( some_nat @ M6 ) )
           => ~ ( ord_less_eq_nat @ M @ M6 ) ) ) ).

% le_some_optE
thf(fact_2271_le__some__optE,axiom,
    ! [M: int,X: option_int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ M ) @ X )
     => ~ ! [M6: int] :
            ( ( X
              = ( some_int @ M6 ) )
           => ~ ( ord_less_eq_int @ M @ M6 ) ) ) ).

% le_some_optE
thf(fact_2272_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_2273_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_2274_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_2275_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_2276_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_2277_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_2278_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_2279_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_2280_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_2281_of__bool__eq__iff,axiom,
    ! [P4: $o,Q2: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P4 )
        = ( zero_n2687167440665602831ol_nat @ Q2 ) )
      = ( P4 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_2282_of__bool__eq__iff,axiom,
    ! [P4: $o,Q2: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P4 )
        = ( zero_n2684676970156552555ol_int @ Q2 ) )
      = ( P4 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_2283_of__bool__eq__iff,axiom,
    ! [P4: $o,Q2: $o] :
      ( ( ( zero_n356916108424825756nteger @ P4 )
        = ( zero_n356916108424825756nteger @ Q2 ) )
      = ( P4 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_2284_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_2285_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_2286_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_2287_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_2288_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_2289_one__reorient,axiom,
    ! [X: assn] :
      ( ( one_one_assn = X )
      = ( X = one_one_assn ) ) ).

% one_reorient
thf(fact_2290_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_2291_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_2292_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_2293_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_2294_power__increasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_2295_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_2296_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_2297_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_2298_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_2299_split__of__bool__asm,axiom,
    ! [P: complex > $o,P4: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_complex ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_complex ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_2300_split__of__bool__asm,axiom,
    ! [P: real > $o,P4: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_real ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_real ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_2301_split__of__bool__asm,axiom,
    ! [P: rat > $o,P4: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_rat ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_2302_split__of__bool__asm,axiom,
    ! [P: nat > $o,P4: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_nat ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_2303_split__of__bool__asm,axiom,
    ! [P: int > $o,P4: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_int ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_2304_split__of__bool__asm,axiom,
    ! [P: code_integer > $o,P4: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_Code_integer ) )
            | ( ~ P4
              & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_2305_split__of__bool,axiom,
    ! [P: complex > $o,P4: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
      = ( ( P4
         => ( P @ one_one_complex ) )
        & ( ~ P4
         => ( P @ zero_zero_complex ) ) ) ) ).

% split_of_bool
thf(fact_2306_split__of__bool,axiom,
    ! [P: real > $o,P4: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
      = ( ( P4
         => ( P @ one_one_real ) )
        & ( ~ P4
         => ( P @ zero_zero_real ) ) ) ) ).

% split_of_bool
thf(fact_2307_split__of__bool,axiom,
    ! [P: rat > $o,P4: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
      = ( ( P4
         => ( P @ one_one_rat ) )
        & ( ~ P4
         => ( P @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_2308_split__of__bool,axiom,
    ! [P: nat > $o,P4: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
      = ( ( P4
         => ( P @ one_one_nat ) )
        & ( ~ P4
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_2309_split__of__bool,axiom,
    ! [P: int > $o,P4: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
      = ( ( P4
         => ( P @ one_one_int ) )
        & ( ~ P4
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_2310_split__of__bool,axiom,
    ! [P: code_integer > $o,P4: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
      = ( ( P4
         => ( P @ one_one_Code_integer ) )
        & ( ~ P4
         => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).

% split_of_bool
thf(fact_2311_of__bool__def,axiom,
    ( zero_n1201886186963655149omplex
    = ( ^ [P3: $o] : ( if_complex @ P3 @ one_one_complex @ zero_zero_complex ) ) ) ).

% of_bool_def
thf(fact_2312_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P3: $o] : ( if_real @ P3 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_2313_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P3: $o] : ( if_rat @ P3 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_2314_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P3: $o] : ( if_nat @ P3 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_2315_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P3: $o] : ( if_int @ P3 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_2316_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P3: $o] : ( if_Code_integer @ P3 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_2317_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ X )
      = ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
thf(fact_2318_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_2319_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N5 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2320_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_2321_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_2322_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_2323_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_2324_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2325_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2326_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2327_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2328_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2329_mult__left__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2330_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_2331_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_2332_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_2333_mult__right__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2334_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_2335_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_2336_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_2337_mult__le__one,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ) ).

% mult_le_one
thf(fact_2338_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_2339_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_2340_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_2341_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_2342_mult__left__le,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2343_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_2344_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_2345_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_2346_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_2347_discrete,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A4: code_integer] : ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A4 @ one_one_Code_integer ) ) ) ) ).

% discrete
thf(fact_2348_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_2349_discrete,axiom,
    ( ord_less_int
    = ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_2350_power__le__one,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer ) ) ) ).

% power_le_one
thf(fact_2351_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_2352_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_2353_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_2354_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_2355_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_2356_mod__power__lem,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( ord_less_eq_nat @ M @ N )
         => ( ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ M ) )
            = zero_zero_int ) )
        & ( ~ ( ord_less_eq_nat @ M @ N )
         => ( ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ M ) )
            = ( power_power_int @ A @ N ) ) ) ) ) ).

% mod_power_lem
thf(fact_2357_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_2358_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_2359_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_2360_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_2361_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_2362_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_2363_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_2364_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).

% signed_take_bit_add
thf(fact_2365_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).

% signed_take_bit_mult
thf(fact_2366_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_2367_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_2368_le__numeral__extra_I3_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% le_numeral_extra(3)
thf(fact_2369_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_2370_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_2371_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_2372_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z6 )
     => ~ ( ord_less_eq_rat @ T @ X4 ) ) ).

% minf(8)
thf(fact_2373_minf_I8_J,axiom,
    ! [T: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z6 )
     => ~ ( ord_le3102999989581377725nteger @ T @ X4 ) ) ).

% minf(8)
thf(fact_2374_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z6 )
     => ~ ( ord_less_eq_real @ T @ X4 ) ) ).

% minf(8)
thf(fact_2375_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z6: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z6 )
     => ~ ( ord_less_eq_num @ T @ X4 ) ) ).

% minf(8)
thf(fact_2376_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z6 )
     => ~ ( ord_less_eq_nat @ T @ X4 ) ) ).

% minf(8)
thf(fact_2377_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z6 )
     => ~ ( ord_less_eq_int @ T @ X4 ) ) ).

% minf(8)
thf(fact_2378_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z6 )
     => ( ord_less_eq_rat @ X4 @ T ) ) ).

% minf(6)
thf(fact_2379_minf_I6_J,axiom,
    ! [T: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z6 )
     => ( ord_le3102999989581377725nteger @ X4 @ T ) ) ).

% minf(6)
thf(fact_2380_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z6 )
     => ( ord_less_eq_real @ X4 @ T ) ) ).

% minf(6)
thf(fact_2381_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z6: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z6 )
     => ( ord_less_eq_num @ X4 @ T ) ) ).

% minf(6)
thf(fact_2382_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z6 )
     => ( ord_less_eq_nat @ X4 @ T ) ) ).

% minf(6)
thf(fact_2383_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z6 )
     => ( ord_less_eq_int @ X4 @ T ) ) ).

% minf(6)
thf(fact_2384_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z6 @ X4 )
     => ( ord_less_eq_rat @ T @ X4 ) ) ).

% pinf(8)
thf(fact_2385_pinf_I8_J,axiom,
    ! [T: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z6 @ X4 )
     => ( ord_le3102999989581377725nteger @ T @ X4 ) ) ).

% pinf(8)
thf(fact_2386_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z6 @ X4 )
     => ( ord_less_eq_real @ T @ X4 ) ) ).

% pinf(8)
thf(fact_2387_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z6: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z6 @ X4 )
     => ( ord_less_eq_num @ T @ X4 ) ) ).

% pinf(8)
thf(fact_2388_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z6 @ X4 )
     => ( ord_less_eq_nat @ T @ X4 ) ) ).

% pinf(8)
thf(fact_2389_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z6 @ X4 )
     => ( ord_less_eq_int @ T @ X4 ) ) ).

% pinf(8)
thf(fact_2390_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z6: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z6 @ X4 )
     => ~ ( ord_less_eq_rat @ X4 @ T ) ) ).

% pinf(6)
thf(fact_2391_pinf_I6_J,axiom,
    ! [T: code_integer] :
    ? [Z6: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z6 @ X4 )
     => ~ ( ord_le3102999989581377725nteger @ X4 @ T ) ) ).

% pinf(6)
thf(fact_2392_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z6: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z6 @ X4 )
     => ~ ( ord_less_eq_real @ X4 @ T ) ) ).

% pinf(6)
thf(fact_2393_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z6: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z6 @ X4 )
     => ~ ( ord_less_eq_num @ X4 @ T ) ) ).

% pinf(6)
thf(fact_2394_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z6: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z6 @ X4 )
     => ~ ( ord_less_eq_nat @ X4 @ T ) ) ).

% pinf(6)
thf(fact_2395_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z6: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z6 @ X4 )
     => ~ ( ord_less_eq_int @ X4 @ T ) ) ).

% pinf(6)
thf(fact_2396_verit__comp__simplify1_I3_J,axiom,
    ! [B4: rat,A5: rat] :
      ( ( ~ ( ord_less_eq_rat @ B4 @ A5 ) )
      = ( ord_less_rat @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2397_verit__comp__simplify1_I3_J,axiom,
    ! [B4: code_integer,A5: code_integer] :
      ( ( ~ ( ord_le3102999989581377725nteger @ B4 @ A5 ) )
      = ( ord_le6747313008572928689nteger @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2398_verit__comp__simplify1_I3_J,axiom,
    ! [B4: real,A5: real] :
      ( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
      = ( ord_less_real @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2399_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A5: num] :
      ( ( ~ ( ord_less_eq_num @ B4 @ A5 ) )
      = ( ord_less_num @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2400_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A5: nat] :
      ( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
      = ( ord_less_nat @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2401_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A5: int] :
      ( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
      = ( ord_less_int @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2402_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_2403_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_2404_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_2405_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_2406_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_2407_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_2408_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_2409_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_2410_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B2: nat] :
        ? [C4: nat] :
          ( B2
          = ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).

% le_iff_add
thf(fact_2411_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_2412_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_2413_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_2414_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_2415_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_2416_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_2417_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_2418_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_2419_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_2420_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_2421_add__mono,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D2 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_2422_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_2423_add__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_2424_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J2 )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2425_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J2 )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2426_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J2 )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2427_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J2 )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2428_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( I = J2 )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2429_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( I = J2 )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2430_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( I = J2 )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2431_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( I = J2 )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2432_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J2 )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2433_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J2 )
        & ( K = L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2434_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J2 )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2435_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J2 )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2436_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_2437_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_2438_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_2439_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_2440_Suc__le__D,axiom,
    ! [N: nat,M7: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
     => ? [M3: nat] :
          ( M7
          = ( suc @ M3 ) ) ) ).

% Suc_le_D
thf(fact_2441_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_2442_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_2443_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_2444_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
             => ( P @ M2 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_2445_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_2446_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R3: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X3: nat] : ( R3 @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z6: nat] :
              ( ( R3 @ X3 @ Y3 )
             => ( ( R3 @ Y3 @ Z6 )
               => ( R3 @ X3 @ Z6 ) ) )
         => ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
           => ( R3 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_2447_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_2448_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_2449_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_2450_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_2451_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_2452_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
thf(fact_2453_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_2454_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_2455_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_2456_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_2457_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_2458_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_2459_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_2460_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_2461_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% less_numeral_extra(4)
thf(fact_2462_exists__leI,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [N6: nat] :
            ( ( ord_less_nat @ N6 @ N )
           => ~ ( P @ N6 ) )
       => ( P @ N ) )
     => ? [N7: nat] :
          ( ( ord_less_eq_nat @ N7 @ N )
          & ( P @ N7 ) ) ) ).

% exists_leI
thf(fact_2463_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
          & ( M5 != N4 ) ) ) ) ).

% nat_less_le
thf(fact_2464_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_2465_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N4: nat] :
          ( ( ord_less_nat @ M5 @ N4 )
          | ( M5 = N4 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_2466_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_2467_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_2468_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J2: nat] :
      ( ! [I3: nat,J: nat] :
          ( ( ord_less_nat @ I3 @ J )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J ) ) )
     => ( ( ord_less_eq_nat @ I @ J2 )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_2469_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2470_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_2471_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_2472_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_2473_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_2474_mult_Ocomm__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.comm_neutral
thf(fact_2475_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_2476_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_2477_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_2478_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_2479_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_2480_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_2481_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_2482_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_2483_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_2484_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_2485_add__le__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).

% add_le_mono
thf(fact_2486_add__le__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% add_le_mono1
thf(fact_2487_trans__le__add1,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).

% trans_le_add1
thf(fact_2488_trans__le__add2,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).

% trans_le_add2
thf(fact_2489_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N4: nat] :
        ? [K3: nat] :
          ( N4
          = ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_2490_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X2: real,Y5: real] :
          ( ( ord_less_real @ X2 @ Y5 )
          | ( X2 = Y5 ) ) ) ) ).

% less_eq_real_def
thf(fact_2491_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_2492_imp__le__cong,axiom,
    ! [X: int,X6: int,P: $o,P2: $o] :
      ( ( X = X6 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
         => ( P = P2 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X6 )
           => P2 ) ) ) ) ).

% imp_le_cong
thf(fact_2493_conj__le__cong,axiom,
    ! [X: int,X6: int,P: $o,P2: $o] :
      ( ( X = X6 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
         => ( P = P2 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X6 )
            & P2 ) ) ) ) ).

% conj_le_cong
thf(fact_2494_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ N )
       => ( ( dvd_dvd_int @ M @ N )
         => ( ( dvd_dvd_int @ N @ M )
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_2495_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).

% zmod_le_nonneg_dividend
thf(fact_2496_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).

% div_le_mono
thf(fact_2497_div__le__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).

% div_le_dividend
thf(fact_2498_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_2499_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_2500_mult__le__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).

% mult_le_mono
thf(fact_2501_mult__le__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).

% mult_le_mono1
thf(fact_2502_mult__le__mono2,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J2 ) ) ) ).

% mult_le_mono2
thf(fact_2503_one__dvd,axiom,
    ! [A: assn] : ( dvd_dvd_assn @ one_one_assn @ A ) ).

% one_dvd
thf(fact_2504_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_2505_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_2506_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_2507_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_2508_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_2509_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_2510_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_2511_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_2512_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).

% mod_less_eq_dividend
thf(fact_2513_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_2514_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_2515_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_2516_i0__lb,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).

% i0_lb
thf(fact_2517_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_2518_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_2519_mult__le__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2520_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_2521_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_2522_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_2523_mult__le__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ C )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2524_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_2525_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_2526_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_2527_mult__le__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2528_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_2529_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_2530_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_2531_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ C )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2532_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_2533_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_2534_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_2535_mult__less__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2536_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_2537_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_2538_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_2539_mult__less__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ C )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2540_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_2541_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_2542_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_2543_mult__less__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2544_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_2545_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_2546_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_2547_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ C )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2548_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_2549_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_2550_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z6: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z6 )
         => ( ( ord_less_rat @ Z6 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z6 @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_2551_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z6: real] :
          ( ( ord_less_real @ zero_zero_real @ Z6 )
         => ( ( ord_less_real @ Z6 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z6 @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_2552_convex__bound__le,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ A )
     => ( ( ord_le3102999989581377725nteger @ Y @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V )
                = one_one_Code_integer )
             => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2553_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_2554_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_2555_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_2556_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_2557_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_2558_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_2559_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_2560_power__Suc__le__self,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2561_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_2562_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_2563_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_2564_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_2565_self__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le3102999989581377725nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2566_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_2567_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_2568_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_2569_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_2570_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_2571_dvd__power__iff,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_2572_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_2573_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_2574_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_2575_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_2576_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_2577_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_2578_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_2579_convex__bound__lt,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ A )
     => ( ( ord_le6747313008572928689nteger @ Y @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V )
                = one_one_Code_integer )
             => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2580_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_2581_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_2582_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_2583_zero__le__numeral,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).

% zero_le_numeral
thf(fact_2584_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_le_numeral
thf(fact_2585_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_le_numeral
thf(fact_2586_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_le_numeral
thf(fact_2587_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_le_numeral
thf(fact_2588_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).

% not_numeral_le_zero
thf(fact_2589_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_2590_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_2591_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_2592_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_2593_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_2594_add__decreasing,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ C @ B )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2595_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_2596_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_2597_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_2598_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_2599_add__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ C )
       => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2600_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_2601_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_2602_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_2603_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_2604_add__decreasing2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ A @ B )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2605_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_2606_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_2607_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_2608_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_2609_add__increasing2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
       => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2610_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_2611_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_2612_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_2613_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_2614_add__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2615_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_2616_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_2617_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_2618_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_2619_add__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_nonpos_nonpos
thf(fact_2620_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_2621_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_2622_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_2623_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_2624_add__nonneg__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ( plus_p5714425477246183910nteger @ X @ Y )
            = zero_z3403309356797280102nteger )
          = ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2625_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_2626_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_2627_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_2628_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_2629_add__nonpos__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ Y @ zero_z3403309356797280102nteger )
       => ( ( ( plus_p5714425477246183910nteger @ X @ Y )
            = zero_z3403309356797280102nteger )
          = ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2630_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_2631_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_2632_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_2633_mult__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_2634_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_2635_mult__mono,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_2636_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_2637_mult__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_2638_mult__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2639_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2640_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2641_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2642_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2643_zero__le__square,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ A ) ) ).

% zero_le_square
thf(fact_2644_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_2645_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_2646_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2647_split__mult__pos__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2648_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_2649_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_2650_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_2651_mult__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2652_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_2653_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_2654_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_2655_mult__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2656_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_2657_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_2658_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_2659_mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2660_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_2661_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_2662_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_2663_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_2664_mult__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2665_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_2666_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_2667_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_2668_mult__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2669_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_2670_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_2671_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_2672_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_2673_mult__le__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2674_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_2675_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_2676_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_2677_split__mult__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) )
     => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ).

% split_mult_neg_le
thf(fact_2678_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_2679_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_2680_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_2681_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_2682_mult__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2683_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_2684_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_2685_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_2686_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_2687_mult__nonneg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2688_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_2689_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_2690_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_2691_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_2692_mult__nonpos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2693_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_2694_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_2695_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_2696_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_2697_mult__nonneg__nonpos2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ B @ A ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2698_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_2699_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_2700_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_2701_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_2702_zero__le__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2703_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_2704_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_2705_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_2706_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2707_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_2708_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_2709_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_2710_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_2711_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J2 )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2712_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: code_integer,J2: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ I @ J2 )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2713_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J2 )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2714_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J2 )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2715_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J2 )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2716_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J2 )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2717_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: code_integer,J2: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I @ J2 )
        & ( ord_le3102999989581377725nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2718_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J2 )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2719_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J2 )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2720_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J2 )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2721_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_2722_add__le__less__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_2723_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D2 )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_2724_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_2725_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_2726_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_2727_add__less__le__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_2728_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D2 )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_2729_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_2730_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_2731_verit__le__mono__div__int,axiom,
    ! [A2: int,B3: int,N: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B3 @ N )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B3 @ N ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_2732_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_2733_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_2734_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_2735_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_2736_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_2737_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_2738_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_2739_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_2740_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_2741_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_2742_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_2743_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_2744_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_2745_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_2746_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_2747_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_2748_power__mono,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).

% power_mono
thf(fact_2749_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_2750_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_2751_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_2752_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_2753_zero__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2754_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_2755_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_2756_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_2757_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_2758_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_2759_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_2760_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_2761_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_2762_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_2763_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2764_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2765_zero__less__one,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one
thf(fact_2766_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_2767_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_2768_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2769_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2770_not__one__less__zero,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_less_zero
thf(fact_2771_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_2772_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_2773_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_2774_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_2775_less__numeral__extra_I1_J,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% less_numeral_extra(1)
thf(fact_2776_nat__compl__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N2 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_compl_induct
thf(fact_2777_nat__compl__induct_H,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N2 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_compl_induct'
thf(fact_2778_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer ) ).

% not_numeral_less_one
thf(fact_2779_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_2780_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_2781_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_2782_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_2783_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_nat @ B @ ( suc @ A ) ) )
      = ( B = A ) ) ).

% nat_in_between_eq(2)
thf(fact_2784_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_nat @ A @ B )
        & ( ord_less_eq_nat @ B @ ( suc @ A ) ) )
      = ( B
        = ( suc @ A ) ) ) ).

% nat_in_between_eq(1)
thf(fact_2785_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_2786_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_2787_dec__induct,axiom,
    ! [I: nat,J2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( P @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J2 )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J2 ) ) ) ) ).

% dec_induct
thf(fact_2788_inc__induct,axiom,
    ! [I: nat,J2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( P @ J2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J2 )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_2789_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_2790_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_2791_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_2792_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_2793_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_2794_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_2795_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_2796_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_2797_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_2798_less__add__one,axiom,
    ! [A: code_integer] : ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) ) ).

% less_add_one
thf(fact_2799_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_2800_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_2801_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_2802_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_2803_add__mono1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ B @ one_one_Code_integer ) ) ) ).

% add_mono1
thf(fact_2804_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_2805_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_2806_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_2807_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_2808_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_2809_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_2810_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_2811_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_2812_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_2813_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_2814_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_2815_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_2816_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_2817_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_2818_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_2819_less__1__mult,axiom,
    ! [M: code_integer,N: code_integer] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ M )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ N )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_2820_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_2821_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_2822_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_2823_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M3: nat,N2: nat] :
          ( ( ord_less_nat @ M3 @ N2 )
         => ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_2824_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2825_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2826_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2827_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2828_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2829_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2830_power__le__dvd,axiom,
    ! [A: nat,N: nat,B: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_2831_power__le__dvd,axiom,
    ! [A: real,N: nat,B: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_2832_power__le__dvd,axiom,
    ! [A: int,N: nat,B: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_2833_power__le__dvd,axiom,
    ! [A: complex,N: nat,B: complex,M: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_2834_power__le__dvd,axiom,
    ! [A: code_integer,N: nat,B: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_2835_power__le__dvd,axiom,
    ! [A: rat,N: nat,B: rat,M: nat] :
      ( ( dvd_dvd_rat @ ( power_power_rat @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_rat @ ( power_power_rat @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_2836_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N: nat,M: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2837_dvd__power__le,axiom,
    ! [X: real,Y: real,N: nat,M: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2838_dvd__power__le,axiom,
    ! [X: int,Y: int,N: nat,M: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2839_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N: nat,M: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2840_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,N: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2841_dvd__power__le,axiom,
    ! [X: rat,Y: rat,N: nat,M: nat] :
      ( ( dvd_dvd_rat @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2842_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).

% Suc_div_le_mono
thf(fact_2843_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = one_one_assn )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N ) @ ( power_power_assn @ Y @ N ) )
        = one_one_assn ) ) ).

% left_right_inverse_power
thf(fact_2844_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_2845_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) )
        = one_one_Code_integer ) ) ).

% left_right_inverse_power
thf(fact_2846_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_2847_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_2848_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_2849_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_2850_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_2851_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_2852_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_2853_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_mult_le_cancel1
thf(fact_2854_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_2855_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_2856_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_2857_power__0,axiom,
    ! [A: assn] :
      ( ( power_power_assn @ A @ zero_zero_nat )
      = one_one_assn ) ).

% power_0
thf(fact_2858_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2859_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2860_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2861_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_2862_power__0,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ zero_zero_nat )
      = one_one_Code_integer ) ).

% power_0
thf(fact_2863_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_2864_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_2865_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_2866_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_2867_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_2868_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_2869_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_2870_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_2871_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_2872_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_2873_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_2874_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_2875_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_2876_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_2877_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_2878_mlex__leI,axiom,
    ! [A: nat,A5: nat,B: nat,B4: nat,N5: nat] :
      ( ( ord_less_eq_nat @ A @ A5 )
     => ( ( ord_less_eq_nat @ B @ B4 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N5 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N5 ) @ B4 ) ) ) ) ).

% mlex_leI
thf(fact_2879_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_2880_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_2881_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_2882_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_2883_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_2884_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_2885_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( ( dvd_dvd_int @ Z @ N )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ Z @ N ) ) ) ).

% zdvd_imp_le
thf(fact_2886_int__mod__ge,axiom,
    ! [A: int,N: int] :
      ( ( ord_less_int @ A @ N )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ A @ ( modulo_modulo_int @ A @ N ) ) ) ) ).

% int_mod_ge
thf(fact_2887_int__mod__eq,axiom,
    ! [B: int,N: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ N )
       => ( ( ( modulo_modulo_int @ A @ N )
            = ( modulo_modulo_int @ B @ N ) )
         => ( ( modulo_modulo_int @ A @ N )
            = B ) ) ) ) ).

% int_mod_eq
thf(fact_2888_int__mod__lem,axiom,
    ! [N: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ N )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ B )
          & ( ord_less_int @ B @ N ) )
        = ( ( modulo_modulo_int @ B @ N )
          = B ) ) ) ).

% int_mod_lem
thf(fact_2889_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_2890_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_2891_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
        & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% neg_mod_conj
thf(fact_2892_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).

% pos_mod_conj
thf(fact_2893_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( modulo_modulo_int @ I @ K )
        = I )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zmod_trivial_iff
thf(fact_2894_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( ( L = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K ) )
        | ( ord_less_int @ zero_zero_int @ L ) ) ) ).

% mod_int_pos_iff
thf(fact_2895_div__mult__le,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ A ) ).

% div_mult_le
thf(fact_2896_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_2897_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_2898_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).

% mod_Suc_le_divisor
thf(fact_2899_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_2900_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_2901_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_2902_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_2903_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_2904_nonneg__mod__div,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 @ ( modulo_modulo_int @ A @ B ) )
          & ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% nonneg_mod_div
thf(fact_2905_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( times_times_nat @ M @ N ) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_2906_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_2907_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_2908_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_2909_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_2910_pow_Osimps_I1_J,axiom,
    ! [X: num] :
      ( ( pow @ X @ one )
      = X ) ).

% pow.simps(1)
thf(fact_2911_emep1,axiom,
    ! [N: int,D2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ D2 )
         => ( ( modulo_modulo_int @ ( plus_plus_int @ N @ one_one_int ) @ D2 )
            = ( plus_plus_int @ ( modulo_modulo_int @ N @ D2 ) @ one_one_int ) ) ) ) ) ).

% emep1
thf(fact_2912_eme1p,axiom,
    ! [N: int,D2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ D2 )
         => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ N ) @ D2 )
            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ N @ D2 ) ) ) ) ) ) ).

% eme1p
thf(fact_2913_verit__le__mono__div,axiom,
    ! [A2: nat,B3: nat,N: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B3 @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B3 @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_2914_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N2: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_2915_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N2: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_2916_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_2917_add__neg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_neg_nonpos
thf(fact_2918_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_2919_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_2920_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_2921_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_2922_add__nonneg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_2923_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_2924_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_2925_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_2926_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_2927_add__nonpos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_nonpos_neg
thf(fact_2928_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_2929_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_2930_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_2931_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_2932_add__pos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_2933_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_2934_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_2935_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_2936_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_2937_add__strict__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2938_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_2939_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_2940_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_2941_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_2942_add__strict__increasing2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2943_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_2944_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_2945_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_2946_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_2947_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_2948_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_2949_mult__le__cancel__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
        = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2950_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_2951_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_2952_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_2953_mult__le__cancel__iff2,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Z @ X ) @ ( times_3573771949741848930nteger @ Z @ Y ) )
        = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2954_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_2955_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_2956_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_2957_mult__le__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2958_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_2959_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_2960_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_2961_mult__le__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2962_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_2963_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_2964_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_2965_mult__left__less__imp__less,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2966_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_2967_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_2968_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_2969_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2970_mult__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2971_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2972_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2973_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2974_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_2975_mult__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2976_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_2977_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_2978_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_2979_mult__right__less__imp__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2980_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_2981_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_2982_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_2983_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2984_mult__strict__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2985_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2986_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2987_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2988_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_2989_mult__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2990_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_2991_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_2992_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_2993_mult__le__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le3102999989581377725nteger @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2994_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_2995_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_2996_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_2997_mult__le__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2998_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_2999_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_3000_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_3001_mult__left__le__imp__le,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3002_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_3003_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_3004_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_3005_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_3006_mult__right__le__imp__le,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3007_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_3008_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_3009_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_3010_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3011_mult__le__less__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3012_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3013_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3014_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3015_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3016_mult__less__le__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3017_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3018_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3019_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( 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 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3020_sum__squares__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3021_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_3022_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_3023_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_3024_sum__squares__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3025_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_3026_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_3027_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_3028_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_3029_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_3030_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_3031_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_3032_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_3033_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_3034_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_3035_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_3036_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_3037_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_3038_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_3039_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_3040_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_3041_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_3042_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_3043_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_3044_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_3045_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3046_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_3047_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_3048_div__positive,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_positive
thf(fact_3049_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_3050_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_3051_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_3052_power__less__imp__less__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3053_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_3054_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_3055_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_3056_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3057_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_3058_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_3059_power__inject__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( suc @ N ) )
        = ( power_8256067586552552935nteger @ B @ ( suc @ N ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3060_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_3061_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_3062_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_3063_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_3064_power__le__imp__le__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) @ ( power_8256067586552552935nteger @ B @ ( suc @ N ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3065_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_3066_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_3067_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_3068_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_3069_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( modulo364778990260209775nteger @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_3070_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_3071_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_3072_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_3073_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_3074_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_3075_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_3076_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_3077_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_3078_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_3079_zero__less__two,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% zero_less_two
thf(fact_3080_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_3081_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_3082_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_3083_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_3084_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_3085_div__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% div_add_self1
thf(fact_3086_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_3087_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_3088_div__add__self2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% div_add_self2
thf(fact_3089_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_3090_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_3091_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_3092_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_3093_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_3094_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_3095_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_3096_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_3097_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_3098_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_3099_power__gt1__lemma,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_3100_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_3101_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_3102_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_3103_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_3104_power__less__power__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_3105_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_3106_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_3107_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_3108_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_3109_power__gt1,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3110_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_3111_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_3112_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_3113_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_3114_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
          = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
          = zero_z3403309356797280102nteger ) ) ) ).

% power_0_left
thf(fact_3115_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_3116_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_3117_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_3118_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_3119_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_3120_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ N @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% div_greater_zero_iff
thf(fact_3121_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_3122_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3123_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3124_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3125_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3126_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3127_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_3128_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_3129_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_3130_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_3131_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_3132_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_3133_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_3134_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_3135_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_3136_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_3137_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_3138_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_3139_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_3140_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_3141_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_3142_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_3143_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_3144_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_3145_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_3146_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_3147_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_3148_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ K @ N ) ) ) ).

% dvd_imp_le
thf(fact_3149_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_3150_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_3151_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_3152_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_3153_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_3154_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_3155_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_le_divisor
thf(fact_3156_int__mod__ge_H,axiom,
    ! [B: int,N: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ ( plus_plus_int @ B @ N ) @ ( modulo_modulo_int @ B @ N ) ) ) ) ).

% int_mod_ge'
thf(fact_3157_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L )
          = ( plus_plus_int @ K @ L ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_3158_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_3159_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q3: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_3160_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N @ Q2 ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ~ ! [S3: nat] :
              ( N
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_3161_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N @ Q2 ) )
     => ( ( ord_less_eq_nat @ N @ M )
       => ~ ! [S3: nat] :
              ( M
             != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_3162_zdiv__le__dividend,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ A ) ) ) ).

% zdiv_le_dividend
thf(fact_3163_zdiv__mono1,axiom,
    ! [A: int,A5: int,B: int] :
      ( ( ord_less_eq_int @ A @ A5 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A5 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_3164_zdiv__mono2,axiom,
    ! [A: int,B4: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B4 )
       => ( ( ord_less_eq_int @ B4 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_3165_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( divide_divide_int @ I @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_3166_zdiv__mono1__neg,axiom,
    ! [A: int,A5: int,B: int] :
      ( ( ord_less_eq_int @ A @ A5 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_3167_zdiv__mono2__neg,axiom,
    ! [A: int,B4: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B4 )
       => ( ( ord_less_eq_int @ B4 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_3168_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
      = ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K )
          & ( ord_less_eq_int @ zero_zero_int @ L ) )
        | ( ( ord_less_int @ K @ zero_zero_int )
          & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_3169_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ L @ K )
     => ( ( ord_less_int @ zero_zero_int @ L )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).

% div_positive_int
thf(fact_3170_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_3171_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_3172_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
        = ( ord_less_eq_int @ K @ I ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_3173_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_3174_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_3175_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ( ord_less_eq_int @ B @ A )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_3176_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_3177_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).

% div_less_dividend
thf(fact_3178_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_3179_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_3180_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_3181_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).

% int_div_less_self
thf(fact_3182_power__2__mult__step__le,axiom,
    ! [N3: nat,N: nat,K4: nat,K: nat] :
      ( ( ord_less_eq_nat @ N3 @ N )
     => ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ K4 ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) )
       => ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( plus_plus_nat @ K4 @ one_one_nat ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).

% power_2_mult_step_le
thf(fact_3183_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_3184_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_3185_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% signed_take_bit_int_less_exp
thf(fact_3186_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_3187_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_3188_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_3189_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_3190_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_3191_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_3192_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_3193_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_3194_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_3195_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_3196_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_3197_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_3198_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_3199_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_3200_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_3201_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_3202_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_3203_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_3204_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_3205_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_3206_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ( power_8256067586552552935nteger @ A @ N )
              = ( power_8256067586552552935nteger @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3207_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_3208_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_3209_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_3210_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_3211_power__eq__imp__eq__base,axiom,
    ! [A: code_integer,N: nat,B: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ N )
        = ( power_8256067586552552935nteger @ B @ N ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3212_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_3213_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_3214_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_3215_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_3216_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_3217_pos__mod__sign2,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% pos_mod_sign2
thf(fact_3218_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_3219_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_3220_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_3221_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_3222_power__Suc__less,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_3223_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).

% self_le_ge2_pow
thf(fact_3224_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_eq_le
thf(fact_3225_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_imp_le
thf(fact_3226_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_3227_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_3228_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_3229_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_3230_power__Suc__less__one,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) @ one_one_Code_integer ) ) ) ).

% power_Suc_less_one
thf(fact_3231_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_3232_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_3233_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_3234_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_3235_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: code_integer] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N5 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3236_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_3237_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_3238_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_3239_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_3240_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_3241_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_3242_one__power2,axiom,
    ( ( power_8256067586552552935nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_power2
thf(fact_3243_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_3244_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B5: code_integer] :
              ( ( B5 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B5 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B5 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B5 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B5 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_3245_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B5: nat] :
              ( ( B5 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B5 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B5 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B5 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B5 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_3246_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B5: int] :
              ( ( B5 != zero_zero_int )
             => ( ( dvd_dvd_int @ B5 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B5 )
                 => ( ( ( divide_divide_int @ one_one_int @ B5 )
                      = A )
                   => ( ( ( times_times_int @ A @ B5 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_3247_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_3248_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_3249_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_3250_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_3251_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_3252_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_3253_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_3254_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_3255_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_3256_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_3257_one__less__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_3258_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
       => ( ( divide_divide_nat @ M @ N )
          = Q2 ) ) ) ).

% div_nat_eqI
thf(fact_3259_td__gal,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ B @ C ) @ A )
        = ( ord_less_eq_nat @ B @ ( divide_divide_nat @ A @ C ) ) ) ) ).

% td_gal
thf(fact_3260_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_3261_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_3262_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_3263_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_3264_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_3265_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_3266_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_3267_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
thf(fact_3268_nmod2,axiom,
    ! [N: int] :
      ( ( ( modulo_modulo_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int )
      | ( ( modulo_modulo_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% nmod2
thf(fact_3269_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_3270_q__pos__lemma,axiom,
    ! [B4: int,Q6: int,R4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q6 ) @ R4 ) )
     => ( ( ord_less_int @ R4 @ B4 )
       => ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( ord_less_eq_int @ zero_zero_int @ Q6 ) ) ) ) ).

% q_pos_lemma
thf(fact_3271_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B4: int,Q6: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q6 ) @ R4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q6 ) @ R4 ) )
       => ( ( ord_less_int @ R4 @ B4 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
           => ( ( ord_less_int @ zero_zero_int @ B4 )
             => ( ( ord_less_eq_int @ B4 @ B )
               => ( ord_less_eq_int @ Q2 @ Q6 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3272_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B4: int,Q6: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q6 ) @ R4 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q6 ) @ R4 ) @ zero_zero_int )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
           => ( ( ord_less_int @ zero_zero_int @ B4 )
             => ( ( ord_less_eq_int @ B4 @ B )
               => ( ord_less_eq_int @ Q6 @ Q2 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3273_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R4: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
       => ( ( ord_less_int @ R4 @ B )
         => ( ( ord_less_int @ R2 @ B )
           => ( ord_less_eq_int @ Q6 @ Q2 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3274_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R4: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( ord_less_int @ B @ R4 )
           => ( ord_less_eq_int @ Q2 @ Q6 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3275_incr__mult__lemma,axiom,
    ! [D2: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D2 ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X4: int] :
              ( ( P @ X4 )
             => ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3276_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ 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_3277_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ 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_3278_split__zmod,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_3279_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_3280_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_3281_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_3282_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_3283_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_3284_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_3285_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_3286_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_3287_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_3288_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_signed_take_bit_iff
thf(fact_3289_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_3290_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_3291_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_3292_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_3293_zero__le__power2,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_3294_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_3295_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_3296_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_3297_power2__eq__imp__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3298_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_3299_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_3300_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_3301_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_3302_power2__le__imp__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3303_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_3304_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_3305_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_3306_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_3307_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_3308_power__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3309_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_3310_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_3311_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_3312_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3313_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_3314_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_3315_power__mono__odd,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_le3102999989581377725nteger @ A @ B )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_3316_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_3317_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_3318_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_3319_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q4: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
            & ( P @ Q4 ) ) ) ) ).

% split_div'
thf(fact_3320_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% dvd_power_iff_le
thf(fact_3321_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_3322_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_3323_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_3324_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_3325_axxmod2,axiom,
    ! [X: int] :
      ( ( ( modulo_modulo_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ X ) @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int )
      & ( ( modulo_modulo_int @ ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ X ) @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% axxmod2
thf(fact_3326_split__zdiv,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( divide_divide_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 ) ) ) ) ) ).

% split_zdiv
thf(fact_3327_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( divide_divide_int @ A @ B )
            = Q2 ) ) ) ) ).

% int_div_neg_eq
thf(fact_3328_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q2 ) ) ) ) ).

% int_div_pos_eq
thf(fact_3329_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I2: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_3330_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I2: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_3331_z1pmod2,axiom,
    ! [B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = one_one_int ) ).

% z1pmod2
thf(fact_3332_p1mod22k,axiom,
    ! [B: int,N: nat] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ one_one_int ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) @ one_one_int ) ) ).

% p1mod22k
thf(fact_3333_p1mod22k_H,axiom,
    ! [B: int,N: nat] :
      ( ( 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 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% p1mod22k'
thf(fact_3334_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_3335_bits__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [A6: nat] :
          ( ( ( divide_divide_nat @ A6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A6 )
         => ( P @ A6 ) )
     => ( ! [A6: nat,B5: $o] :
            ( ( P @ A6 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A6 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A6 )
             => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A6 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_3336_bits__induct,axiom,
    ! [P: int > $o,A: int] :
      ( ! [A6: int] :
          ( ( ( divide_divide_int @ A6 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A6 )
         => ( P @ A6 ) )
     => ( ! [A6: int,B5: $o] :
            ( ( P @ A6 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A6 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A6 )
             => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A6 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_3337_bits__induct,axiom,
    ! [P: code_integer > $o,A: code_integer] :
      ( ! [A6: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A6 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A6 )
         => ( P @ A6 ) )
     => ( ! [A6: code_integer,B5: $o] :
            ( ( P @ A6 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A6 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A6 )
             => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A6 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_3338_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_3339_power2__less__imp__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3340_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_3341_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_3342_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_3343_sum__power2__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3344_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_3345_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_3346_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_3347_sum__power2__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3348_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_3349_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_3350_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_3351_zero__le__even__power_H,axiom,
    ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_3352_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_3353_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_3354_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_3355_zero__le__power__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_3356_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_3357_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_3358_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_3359_zero__le__odd__power,axiom,
    ! [N: nat,A: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N ) )
        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ) ).

% zero_le_odd_power
thf(fact_3360_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_3361_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_3362_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_3363_zero__le__even__power,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_3364_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_3365_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_3366_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_3367_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_3368_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_3369_two__pow__div__gt__le,axiom,
    ! [V: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ V @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% two_pow_div_gt_le
thf(fact_3370_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_3371_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_3372_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_3373_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_3374_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_3375_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_3376_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D2: 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 @ D2 ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D2 @ ( 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_3377_axxdiv2,axiom,
    ! [X: int] :
      ( ( ( divide_divide_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ X ) @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = X )
      & ( ( divide_divide_int @ ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ X ) @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = X ) ) ).

% axxdiv2
thf(fact_3378_z1pdiv2,axiom,
    ! [B: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = B ) ).

% z1pdiv2
thf(fact_3379_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_3380_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_3381_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_3382_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_3383_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_3384_odd__0__le__power__imp__0__le,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3385_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_3386_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_3387_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_3388_sb__inc__lem,axiom,
    ! [A: int,K: nat] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) ) ) ).

% sb_inc_lem
thf(fact_3389_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X @ M ) )
          | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3390_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X @ M ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3391_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X @ M ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3392_power__le__zero__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ zero_z3403309356797280102nteger )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_3393_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_3394_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_3395_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_3396_nat__div__eq__Suc__0__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ( divide_divide_nat @ N @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_div_eq_Suc_0_iff
thf(fact_3397_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_3398_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_3399_atLeastLessThan__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or4029947393144176647an_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I )
        & ( ord_less_rat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_3400_atLeastLessThan__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( ( member_real @ I @ ( set_or66887138388493659n_real @ L @ U ) )
      = ( ( ord_less_eq_real @ L @ I )
        & ( ord_less_real @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_3401_atLeastLessThan__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I )
        & ( ord_less_set_nat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_3402_atLeastLessThan__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( ( member_num @ I @ ( set_or1222409239386451017an_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I )
        & ( ord_less_num @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_3403_atLeastLessThan__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I )
        & ( ord_less_nat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_3404_atLeastLessThan__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I )
        & ( ord_less_int @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_3405_atLeastLessThan__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( ( member_Code_integer @ I @ ( set_or8404916559141939852nteger @ L @ U ) )
      = ( ( ord_le3102999989581377725nteger @ L @ I )
        & ( ord_le6747313008572928689nteger @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_3406_ivl__subset,axiom,
    ! [I: real,J2: real,M: real,N: real] :
      ( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I @ J2 ) @ ( set_or66887138388493659n_real @ M @ N ) )
      = ( ( ord_less_eq_real @ J2 @ I )
        | ( ( ord_less_eq_real @ M @ I )
          & ( ord_less_eq_real @ J2 @ N ) ) ) ) ).

% ivl_subset
thf(fact_3407_ivl__subset,axiom,
    ! [I: num,J2: num,M: num,N: num] :
      ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ I @ J2 ) @ ( set_or1222409239386451017an_num @ M @ N ) )
      = ( ( ord_less_eq_num @ J2 @ I )
        | ( ( ord_less_eq_num @ M @ I )
          & ( ord_less_eq_num @ J2 @ N ) ) ) ) ).

% ivl_subset
thf(fact_3408_ivl__subset,axiom,
    ! [I: nat,J2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J2 ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ J2 @ I )
        | ( ( ord_less_eq_nat @ M @ I )
          & ( ord_less_eq_nat @ J2 @ N ) ) ) ) ).

% ivl_subset
thf(fact_3409_ivl__subset,axiom,
    ! [I: int,J2: int,M: int,N: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J2 ) @ ( set_or4662586982721622107an_int @ M @ N ) )
      = ( ( ord_less_eq_int @ J2 @ I )
        | ( ( ord_less_eq_int @ M @ I )
          & ( ord_less_eq_int @ J2 @ N ) ) ) ) ).

% ivl_subset
thf(fact_3410_ivl__subset,axiom,
    ! [I: code_integer,J2: code_integer,M: code_integer,N: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or8404916559141939852nteger @ I @ J2 ) @ ( set_or8404916559141939852nteger @ M @ N ) )
      = ( ( ord_le3102999989581377725nteger @ J2 @ I )
        | ( ( ord_le3102999989581377725nteger @ M @ I )
          & ( ord_le3102999989581377725nteger @ J2 @ N ) ) ) ) ).

% ivl_subset
thf(fact_3411_T__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ord_less_eq_nat @ ( vEBT_V441764108873111860ildupi @ N ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% T_vebt_buildupi_univ
thf(fact_3412_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_real @ zero_zero_real @ ( vEBT_VEBT_cnt @ T ) ) ).

% cnt_non_neg
thf(fact_3413_pos__mult__pos__ge,axiom,
    ! [X: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ ( times_times_int @ N @ one_one_int ) @ ( times_times_int @ N @ X ) ) ) ) ).

% pos_mult_pos_ge
thf(fact_3414_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3415_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3416_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3417_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3418_two__realpow__ge__two,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).

% two_realpow_ge_two
thf(fact_3419_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3420_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: 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 ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ 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 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3421_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: 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 ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ 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 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3422_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(90)
thf(fact_3423_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_3424_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_3425_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_3426_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = ( semiri8010041392384452111omplex @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_3427_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_3428_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3429_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_3430_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_3431_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_3432_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_3433_diff__zero,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_zero
thf(fact_3434_diff__zero,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% diff_zero
thf(fact_3435_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_3436_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_3437_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_3438_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_3439_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_3440_diff__0__right,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_0_right
thf(fact_3441_diff__0__right,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% diff_0_right
thf(fact_3442_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_3443_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_3444_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_3445_diff__self,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% diff_self
thf(fact_3446_diff__self,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% diff_self
thf(fact_3447_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_3448_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_3449_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_3450_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3451_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3452_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3453_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3454_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3455_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3456_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_3457_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_3458_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_3459_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_3460_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_3461_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_3462_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_3463_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_3464_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_3465_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_3466_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_3467_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_3468_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_3469_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_3470_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_3471_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_3472_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N ) ) ).

% semiring_norm(88)
thf(fact_3473_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N ) ) ).

% semiring_norm(89)
thf(fact_3474_semiring__norm_I84_J,axiom,
    ! [N: num] :
      ( one
     != ( bit1 @ N ) ) ).

% semiring_norm(84)
thf(fact_3475_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_3476_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_3477_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_3478_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_3479_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mod_self2
thf(fact_3480_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_3481_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_3482_diff__diff__left,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% diff_diff_left
thf(fact_3483_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_3484_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(73)
thf(fact_3485_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(80)
thf(fact_3486_dbl__simps_I2_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% dbl_simps(2)
thf(fact_3487_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_3488_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_3489_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_3490_dbl__simps_I2_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% dbl_simps(2)
thf(fact_3491_diff__ge__0__iff__ge,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_3492_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_3493_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_3494_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_3495_zero__comp__diff__simps_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_3496_zero__comp__diff__simps_I1_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_3497_zero__comp__diff__simps_I1_J,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_eq_real @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_3498_zero__comp__diff__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_3499_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_3500_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_3501_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_3502_diff__gt__0__iff__gt,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_3503_zero__comp__diff__simps_I2_J,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_real @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_3504_zero__comp__diff__simps_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_3505_zero__comp__diff__simps_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_3506_zero__comp__diff__simps_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_3507_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_3508_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_3509_diff__numeral__special_I9_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(9)
thf(fact_3510_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_3511_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_3512_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_3513_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_3514_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_3515_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_3516_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_3517_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_3518_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_3519_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_3520_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_3521_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_3522_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_3523_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_3524_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_3525_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_3526_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_3527_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_3528_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_3529_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri681578069525770553at_rat @ M )
        = zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3530_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3531_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3532_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3533_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3534_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3535_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_3536_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_z3403309356797280102nteger
        = ( semiri4939895301339042750nteger @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_3537_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_3538_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_3539_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_3540_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_3541_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% semiring_1_class.of_nat_0
thf(fact_3542_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri4939895301339042750nteger @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% semiring_1_class.of_nat_0
thf(fact_3543_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% semiring_1_class.of_nat_0
thf(fact_3544_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% semiring_1_class.of_nat_0
thf(fact_3545_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% semiring_1_class.of_nat_0
thf(fact_3546_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% semiring_1_class.of_nat_0
thf(fact_3547_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% of_nat_numeral
thf(fact_3548_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% of_nat_numeral
thf(fact_3549_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% of_nat_numeral
thf(fact_3550_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% of_nat_numeral
thf(fact_3551_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
      = ( numera6690914467698888265omplex @ N ) ) ).

% of_nat_numeral
thf(fact_3552_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_3553_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_3554_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_3555_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_3556_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_3557_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_3558_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_3559_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_3560_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_3561_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_add
thf(fact_3562_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_add
thf(fact_3563_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_3564_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_3565_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% of_nat_add
thf(fact_3566_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_3567_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_3568_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_3569_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_3570_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri8010041392384452111omplex @ N )
        = one_one_complex )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3571_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_3572_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_3573_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_3574_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_3575_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3576_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_3577_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_3578_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_3579_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_3580_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_3581_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_3582_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_3583_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_3584_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_3585_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_mult
thf(fact_3586_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_mult
thf(fact_3587_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_3588_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_3589_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% of_nat_mult
thf(fact_3590_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).

% Nat.diff_diff_right
thf(fact_3591_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_3592_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_3593_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri4939895301339042750nteger @ X )
        = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3594_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_3595_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_3596_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_3597_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_3598_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_3599_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W )
        = ( semiri4939895301339042750nteger @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3600_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_3601_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_3602_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_3603_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_3604_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_3605_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( power_power_nat @ M @ N ) )
      = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ M ) @ N ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3606_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
      = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3607_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3608_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3609_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3610_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
      = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3611_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_3612_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_3613_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_3614_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_3615_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(9)
thf(fact_3616_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(7)
thf(fact_3617_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).

% semiring_norm(15)
thf(fact_3618_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).

% semiring_norm(14)
thf(fact_3619_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(72)
thf(fact_3620_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(81)
thf(fact_3621_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_3622_semiring__norm_I77_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).

% semiring_norm(77)
thf(fact_3623_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_3624_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n1201886186963655149omplex @ P ) ) ).

% of_nat_of_bool
thf(fact_3625_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_3626_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_3627_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_nat_of_bool
thf(fact_3628_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_3629_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_3630_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_3631_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_3632_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3633_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3634_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3635_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3636_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3637_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3638_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_3639_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_3640_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3641_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_3642_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_3643_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_3644_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J2 ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_3645_Suc__diff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N @ M ) )
          = ( minus_minus_nat @ N @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_diff
thf(fact_3646_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_3647_semiring__norm_I3_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N ) )
      = ( bit1 @ N ) ) ).

% semiring_norm(3)
thf(fact_3648_semiring__norm_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).

% semiring_norm(4)
thf(fact_3649_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_3650_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_3651_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_3652_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_3653_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(79)
thf(fact_3654_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(74)
thf(fact_3655_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_3656_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_3657_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_3658_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_3659_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_3660_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_3661_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri4939895301339042750nteger @ Y )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_3662_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_3663_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_3664_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_3665_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_3666_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_3667_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N )
        = ( semiri4939895301339042750nteger @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_3668_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_3669_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_3670_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_3671_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_3672_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_3673_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_3674_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3675_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_3676_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_3677_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_3678_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_3679_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3680_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_3681_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_3682_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_3683_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3684_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_3685_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_3686_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_3687_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_3688_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3689_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_3690_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_3691_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_3692_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_3693_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_3694_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_3695_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_3696_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_3697_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_3698_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3699_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_3700_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_3701_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_3702_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_3703_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_3704_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_3705_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_3706_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3707_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_3708_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_3709_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_3710_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_3711_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3712_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_3713_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_3714_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_3715_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_3716_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3717_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_3718_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_3719_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_3720_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_3721_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3722_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_3723_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_3724_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_3725_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_3726_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_3727_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_3728_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_3729_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% even_diff_nat
thf(fact_3730_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_3731_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_3732_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_3733_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_3734_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_3735_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_3736_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X4: real] : ( member_real @ X4 @ S2 )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( member_real @ X3 @ S2 )
           => ( ord_less_eq_real @ X3 @ Z5 ) )
       => ? [Y3: real] :
            ( ! [X4: real] :
                ( ( member_real @ X4 @ S2 )
               => ( ord_less_eq_real @ X4 @ Y3 ) )
            & ! [Z5: real] :
                ( ! [X3: real] :
                    ( ( member_real @ X3 @ S2 )
                   => ( ord_less_eq_real @ X3 @ Z5 ) )
               => ( ord_less_eq_real @ Y3 @ Z5 ) ) ) ) ) ).

% complete_real
thf(fact_3737_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_3738_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% of_nat_diff
thf(fact_3739_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% of_nat_diff
thf(fact_3740_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_3741_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).

% of_nat_diff
thf(fact_3742_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_3743_le__trans,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_eq_nat @ J2 @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_3744_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_3745_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_3746_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_3747_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_3748_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_3749_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_3750_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_3751_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_3752_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_3753_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_3754_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y4: nat] :
                ( ( P @ Y4 )
               => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_3755_cancel__ab__semigroup__add__class_Odiff__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 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_3756_cancel__ab__semigroup__add__class_Odiff__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 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_3757_cancel__ab__semigroup__add__class_Odiff__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 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_3758_cancel__ab__semigroup__add__class_Odiff__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 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_3759_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D2 ) )
     => ( ( A = B )
        = ( C = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_3760_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D2 ) )
     => ( ( A = B )
        = ( C = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_3761_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( A = B )
        = ( C = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_3762_diff__commute,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J2 ) ) ).

% diff_commute
thf(fact_3763_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3764_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3765_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3766_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_3767_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_3768_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_3769_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_3770_mult__of__nat__commute,axiom,
    ! [X: nat,Y: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
      = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_3771_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: complex,Z3: complex] : Y6 = Z3 )
    = ( ^ [A4: complex,B2: complex] :
          ( ( minus_minus_complex @ A4 @ B2 )
          = zero_zero_complex ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3772_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: code_integer,Z3: code_integer] : Y6 = Z3 )
    = ( ^ [A4: code_integer,B2: code_integer] :
          ( ( minus_8373710615458151222nteger @ A4 @ B2 )
          = zero_z3403309356797280102nteger ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3773_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: real,Z3: real] : Y6 = Z3 )
    = ( ^ [A4: real,B2: real] :
          ( ( minus_minus_real @ A4 @ B2 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3774_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: rat,Z3: rat] : Y6 = Z3 )
    = ( ^ [A4: rat,B2: rat] :
          ( ( minus_minus_rat @ A4 @ B2 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3775_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: int,Z3: int] : Y6 = Z3 )
    = ( ^ [A4: int,B2: int] :
          ( ( minus_minus_int @ A4 @ B2 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3776_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D2 ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_3777_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D2 ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_3778_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_3779_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_3780_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_3781_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_3782_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_3783_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_3784_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_3785_diff__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D2 @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).

% diff_mono
thf(fact_3786_diff__mono,axiom,
    ! [A: real,B: real,D2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D2 @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).

% diff_mono
thf(fact_3787_diff__mono,axiom,
    ! [A: int,B: int,D2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D2 @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_mono
thf(fact_3788_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_3789_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_3790_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_3791_diff__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_3792_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_3793_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_3794_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_3795_diff__strict__left__mono,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ C @ A ) @ ( minus_8373710615458151222nteger @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_3796_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D2 ) )
     => ( ( ord_less_real @ A @ B )
        = ( ord_less_real @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_3797_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D2 ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_3798_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_3799_diff__eq__diff__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( minus_8373710615458151222nteger @ A @ B )
        = ( minus_8373710615458151222nteger @ C @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
        = ( ord_le6747313008572928689nteger @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_3800_diff__strict__mono,axiom,
    ! [A: real,B: real,D2: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ D2 @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_3801_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D2 @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_3802_diff__strict__mono,axiom,
    ! [A: int,B: int,D2: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D2 @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_3803_diff__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,D2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ D2 @ C )
       => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ ( minus_8373710615458151222nteger @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_3804_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) )
      = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D2 ) ) ) ).

% add_diff_add
thf(fact_3805_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) )
      = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D2 ) ) ) ).

% add_diff_add
thf(fact_3806_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) )
      = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D2 ) ) ) ).

% add_diff_add
thf(fact_3807_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3808_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3809_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3810_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_3811_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_3812_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_3813_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_3814_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_3815_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_3816_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_3817_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_3818_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_3819_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_3820_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_3821_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_3822_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_3823_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_3824_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_3825_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_3826_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_3827_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_3828_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_3829_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_3830_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_3831_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_3832_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_3833_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_3834_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_3835_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_3836_inf__period_I1_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X3: real,K2: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
     => ( ! [X3: real,K2: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
       => ! [X4: real,K5: real] :
            ( ( ( P @ X4 )
              & ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K5 @ D4 ) ) )
              & ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K5 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3837_inf__period_I1_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X3: rat,K2: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
     => ( ! [X3: rat,K2: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
       => ! [X4: rat,K5: rat] :
            ( ( ( P @ X4 )
              & ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K5 @ D4 ) ) )
              & ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K5 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3838_inf__period_I1_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
       => ! [X4: int,K5: int] :
            ( ( ( P @ X4 )
              & ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D4 ) ) )
              & ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3839_inf__period_I2_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X3: real,K2: real] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
     => ( ! [X3: real,K2: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
       => ! [X4: real,K5: real] :
            ( ( ( P @ X4 )
              | ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K5 @ D4 ) ) )
              | ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K5 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3840_inf__period_I2_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X3: rat,K2: rat] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
     => ( ! [X3: rat,K2: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
       => ! [X4: rat,K5: rat] :
            ( ( ( P @ X4 )
              | ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K5 @ D4 ) ) )
              | ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K5 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3841_inf__period_I2_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P @ X3 )
          = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
       => ! [X4: int,K5: int] :
            ( ( ( P @ X4 )
              | ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D4 ) ) )
              | ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3842_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_3843_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_3844_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_3845_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_3846_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_3847_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_3848_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_3849_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_3850_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_3851_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_3852_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_3853_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_3854_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_3855_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_3856_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N4: nat,M5: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N4 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).

% nat_less_real_le
thf(fact_3857_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_3858_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_3859_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_3860_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N4: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_3861_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] :
      ( ( bit0 @ X22 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_3862_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_3863_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_3864_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_3865_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_3866_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_3867_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_3868_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_3869_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_3870_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_3871_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_3872_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_3873_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_3874_mod__diff__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B4: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A5 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( minus_minus_int @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_3875_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A5 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_3876_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_3877_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_eq
thf(fact_3878_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_3879_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_3880_diff__diff__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ M @ ( minus_minus_nat @ M @ N ) ) )
      = ( ( ord_less_nat @ I @ M )
        & ( ord_less_nat @ I @ N ) ) ) ).

% diff_diff_less
thf(fact_3881_less__imp__diff__less,axiom,
    ! [J2: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J2 @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_3882_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_3883_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_3884_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_3885_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_3886_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_3887_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_3888_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_3889_le__diff__conv,axiom,
    ! [J2: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
      = ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_3890_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).

% Nat.le_diff_conv2
thf(fact_3891_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_3892_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_3893_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ( minus_minus_nat @ J2 @ I )
          = K )
        = ( J2
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_3894_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_3895_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% diff_mult_distrib
thf(fact_3896_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% dvd_diff_nat
thf(fact_3897_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( dvd_dvd_nat @ M @ N )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_3898_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ N ) ) ) ) ).

% dvd_diffD1
thf(fact_3899_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_3900_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% le_mod_geq
thf(fact_3901_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_3902_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_3903_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_3904_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_3905_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) ) ).

% of_nat_0_le_iff
thf(fact_3906_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_3907_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_3908_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_3909_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_3910_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger ) ).

% of_nat_less_0_iff
thf(fact_3911_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_3912_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_3913_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_3914_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
     != zero_zero_rat ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3915_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ N ) )
     != zero_z3403309356797280102nteger ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3916_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
     != zero_zero_real ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3917_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3918_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3919_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
     != zero_zero_complex ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3920_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_3921_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_3922_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_3923_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_3924_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_3925_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_3926_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_3927_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_3928_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_3929_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_3930_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% div_mult2_eq'
thf(fact_3931_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% div_mult2_eq'
thf(fact_3932_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_rat @ ( semiri681578069525770553at_rat @ A ) @ ( semiri681578069525770553at_rat @ B ) )
      = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_3933_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_real @ ( semiri5074537144036343181t_real @ A ) @ ( semiri5074537144036343181t_real @ B ) )
      = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_3934_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
      = ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_3935_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ A ) @ ( semiri1316708129612266289at_nat @ B ) )
      = ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_3936_Abs__fnat__hom__add,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_complex @ ( semiri8010041392384452111omplex @ A ) @ ( semiri8010041392384452111omplex @ B ) )
      = ( semiri8010041392384452111omplex @ ( plus_plus_nat @ A @ B ) ) ) ).

% Abs_fnat_hom_add
thf(fact_3937_le__iff__diff__le__0,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A4: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A4 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% le_iff_diff_le_0
thf(fact_3938_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B2 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_3939_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B2 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_3940_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B2 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_3941_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B2 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_3942_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B2 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_3943_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B2 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_3944_less__iff__diff__less__0,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A4: code_integer,B2: code_integer] : ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A4 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% less_iff_diff_less_0
thf(fact_3945_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_3946_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_3947_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_3948_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_3949_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_3950_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_3951_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_3952_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_3953_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_3954_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_3955_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_3956_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_3957_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_3958_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_3959_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_3960_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_3961_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3962_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3963_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3964_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3965_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J2 @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
         => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J2 @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3966_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J2 @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
         => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J2 @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3967_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3968_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J2 @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J2 @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3969_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_3970_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_3971_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_3972_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_3973_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( plus_p5714425477246183910nteger @ B @ ( minus_8373710615458151222nteger @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_3974_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_3975_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_3976_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_3977_less__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
      = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_3978_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_3979_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_3980_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_3981_diff__less__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
      = ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_3982_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_3983_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_3984_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_3985_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_3986_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_3987_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_3988_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D2: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_3989_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_3990_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_3991_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D2: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_3992_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_3993_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_3994_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_3995_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_3996_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] :
            ( Y
           != ( bit0 @ X23 ) )
       => ~ ! [X33: num] :
              ( Y
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_3997_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 ) ) )
         => ( ! [M3: num] :
                ( X
               != ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) )
           => ( ! [M3: num,N2: num] :
                  ( X
                 != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N2 ) ) )
             => ( ! [M3: num,N2: num] :
                    ( X
                   != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N2 ) ) )
               => ( ! [M3: num] :
                      ( X
                     != ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) )
                 => ( ! [M3: num,N2: num] :
                        ( X
                       != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N2 ) ) )
                   => ~ ! [M3: num,N2: num] :
                          ( X
                         != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_3998_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% of_nat_dvd_iff
thf(fact_3999_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% of_nat_dvd_iff
thf(fact_4000_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_4001_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_4002_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_4003_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_4004_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_4005_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_mod
thf(fact_4006_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mod
thf(fact_4007_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mod
thf(fact_4008_Suc__to__right,axiom,
    ! [N: nat,M: nat] :
      ( ( ( suc @ N )
        = M )
     => ( N
        = ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_to_right
thf(fact_4009_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_4010_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_4011_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N @ M ) )
          = ( minus_minus_nat @ N @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_n_minus_m_eq
thf(fact_4012_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_4013_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_4014_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_4015_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_4016_less__diff__conv,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ).

% less_diff_conv
thf(fact_4017_less__diff__conv2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
        = ( ord_less_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_4018_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X2: real] : ( plus_plus_real @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_4019_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X2: rat] : ( plus_plus_rat @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_4020_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_4021_nat__eq__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J2 @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_4022_nat__eq__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_4023_nat__le__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J2 @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_4024_nat__le__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_4025_nat__diff__add__eq1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J2 @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_4026_nat__diff__add__eq2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_4027_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
      = ( ( ord_less_nat @ N @ M )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_minus_self
thf(fact_4028_nat__minus__mod,axiom,
    ! [N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( minus_minus_nat @ N @ ( modulo_modulo_nat @ N @ M ) ) @ M )
      = zero_zero_nat ) ).

% nat_minus_mod
thf(fact_4029_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% mod_geq
thf(fact_4030_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( ord_less_nat @ M5 @ N4 ) @ M5 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ).

% mod_if
thf(fact_4031_mod__nat__sub,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( modulo_modulo_nat @ ( minus_minus_nat @ X @ Y ) @ Z )
        = ( minus_minus_nat @ X @ Y ) ) ) ).

% mod_nat_sub
thf(fact_4032_nat__minus__mod__plus__right,axiom,
    ! [N: nat,X: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ X ) @ ( modulo_modulo_nat @ N @ M ) ) @ M )
      = ( modulo_modulo_nat @ X @ M ) ) ).

% nat_minus_mod_plus_right
thf(fact_4033_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
          = D3 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
          = D3 ) ) ) ).

% bezout1_nat
thf(fact_4034_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q2 )
          = ( modulo_modulo_nat @ N @ Q2 ) )
        = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_4035_small__powers__of__2,axiom,
    ! [X: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ X )
     => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ X @ one_one_nat ) ) ) ) ).

% small_powers_of_2
thf(fact_4036_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri681578069525770553at_rat @ K )
       != zero_zero_rat )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4037_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri4939895301339042750nteger @ K )
       != zero_z3403309356797280102nteger )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4038_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri5074537144036343181t_real @ K )
       != zero_zero_real )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4039_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri1314217659103216013at_int @ K )
       != zero_zero_int )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4040_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri1316708129612266289at_nat @ K )
       != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4041_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri8010041392384452111omplex @ K )
       != zero_zero_complex )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_4042_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_4043_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_4044_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_4045_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D2 ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_4046_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D2 ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_4047_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_4048_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_4049_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_4050_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_4051_less__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( ord_le6747313008572928689nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_4052_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D2 ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_4053_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_4054_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_4055_less__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_4056_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_4057_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_4058_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_4059_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_4060_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_4061_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_4062_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_4063_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_4064_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_4065_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_4066_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_4067_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_4068_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_4069_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_4070_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_4071_inf__period_I4_J,axiom,
    ! [D2: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D2 @ D4 )
     => ! [X4: real,K5: real] :
          ( ( ~ ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D2 @ ( plus_plus_real @ ( minus_minus_real @ X4 @ ( times_times_real @ K5 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4072_inf__period_I4_J,axiom,
    ! [D2: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D2 @ D4 )
     => ! [X4: rat,K5: rat] :
          ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K5 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4073_inf__period_I4_J,axiom,
    ! [D2: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D4 )
     => ! [X4: int,K5: int] :
          ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4074_inf__period_I3_J,axiom,
    ! [D2: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D2 @ D4 )
     => ! [X4: real,K5: real] :
          ( ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X4 @ T ) )
          = ( dvd_dvd_real @ D2 @ ( plus_plus_real @ ( minus_minus_real @ X4 @ ( times_times_real @ K5 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4075_inf__period_I3_J,axiom,
    ! [D2: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D2 @ D4 )
     => ! [X4: rat,K5: rat] :
          ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ T ) )
          = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K5 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4076_inf__period_I3_J,axiom,
    ! [D2: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D4 )
     => ! [X4: int,K5: int] :
          ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T ) )
          = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4077_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_4078_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_4079_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4080_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_4081_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_4082_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4083_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_4084_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_4085_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4086_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_4087_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_4088_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4089_zmde,axiom,
    ! [B: int,A: int] :
      ( ( times_times_int @ B @ ( divide_divide_int @ A @ B ) )
      = ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% zmde
thf(fact_4090_zmde,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% zmde
thf(fact_4091_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_4092_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_4093_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_4094_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_4095_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_4096_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_4097_power__diff,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_4098_power__diff,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_4099_power__diff,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_4100_power__diff,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_4101_power__diff,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_4102_power__diff,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_4103_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_4104_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_4105_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_4106_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_4107_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_4108_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_4109_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_4110_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_4111_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_4112_reals__Archimedean3,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ! [Y4: real] :
        ? [N2: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).

% reals_Archimedean3
thf(fact_4113_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_4114_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_4115_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_4116_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ K @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ K @ ( suc @ zero_zero_nat ) ) @ M ) ) ) ).

% nz_le_conv_less
thf(fact_4117_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).

% add_eq_if
thf(fact_4118_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 ) )
            | ? [D: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D ) )
                & ~ ( P @ D ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_4119_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 ) )
        & ! [D: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D ) )
           => ( P @ D ) ) ) ) ).

% nat_diff_split
thf(fact_4120_real__of__nat__div,axiom,
    ! [D2: nat,N: nat] :
      ( ( dvd_dvd_nat @ D2 @ N )
     => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D2 ) )
        = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D2 ) ) ) ) ).

% real_of_nat_div
thf(fact_4121_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).

% mult_eq_if
thf(fact_4122_real__of__nat__div__aux,axiom,
    ! [X: nat,D2: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D2 ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D2 ) ) @ ( semiri5074537144036343181t_real @ D2 ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_4123_nat__less__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_4124_nat__less__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J2 @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_4125_mod__nat__add,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ( ( ord_less_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
           => ( ( modulo_modulo_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
              = ( plus_plus_nat @ X @ Y ) ) )
          & ( ~ ( ord_less_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
           => ( ( modulo_modulo_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
              = ( minus_minus_nat @ ( plus_plus_nat @ X @ Y ) @ Z ) ) ) ) ) ) ).

% mod_nat_add
thf(fact_4126_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R2: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q2 @ N )
     => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R2 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q2 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_4127_diff__mod__le,axiom,
    ! [A: nat,D2: nat,B: nat] :
      ( ( ord_less_nat @ A @ D2 )
     => ( ( dvd_dvd_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) @ ( minus_minus_nat @ D2 @ B ) ) ) ) ).

% diff_mod_le
thf(fact_4128_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ R2 @ N )
     => ( ( ord_less_eq_nat @ R2 @ M )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
         => ( ( modulo_modulo_nat @ M @ N )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_4129_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M5: nat,N4: nat] : ( minus_minus_nat @ M5 @ ( times_times_nat @ ( divide_divide_nat @ M5 @ N4 ) @ N4 ) ) ) ) ).

% modulo_nat_def
thf(fact_4130_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc3368934014287244435at_num] :
      ~ ! [F3: nat > num > num,A6: nat,B5: nat,Acc: num] :
          ( X
         != ( produc851828971589881931at_num @ F3 @ ( produc1195630363706982562at_num @ A6 @ ( product_Pair_nat_num @ B5 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_4131_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F3: nat > nat > nat,A6: nat,B5: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A6 @ ( product_Pair_nat_nat @ B5 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_4132_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_4133_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_4134_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_4135_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_4136_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_4137_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_4138_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_4139_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_4140_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_4141_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_4142_power2__commute,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_4143_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_4144_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_4145_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_4146_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_4147_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_4148_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_4149_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M3 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X ) @ C ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_4150_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_4151_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_4152_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_4153_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_4154_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_4155_power__diff__power__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
            = ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) )
            = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_4156_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_4157_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_4158_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_4159_power3__eq__cube,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_4160_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_4161_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_4162_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_4163_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_4164_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_4165_power__eq__if,axiom,
    ( power_power_assn
    = ( ^ [P3: assn,M5: nat] : ( if_assn @ ( M5 = zero_zero_nat ) @ one_one_assn @ ( times_times_assn @ P3 @ ( power_power_assn @ P3 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4166_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P3: complex,M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P3 @ ( power_power_complex @ P3 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4167_power__eq__if,axiom,
    ( power_8256067586552552935nteger
    = ( ^ [P3: code_integer,M5: nat] : ( if_Code_integer @ ( M5 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P3 @ ( power_8256067586552552935nteger @ P3 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4168_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P3: real,M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P3 @ ( power_power_real @ P3 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4169_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P3: rat,M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P3 @ ( power_power_rat @ P3 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4170_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P3: nat,M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4171_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P3: int,M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P3 @ ( power_power_int @ P3 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4172_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_4173_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_4174_power__minus__mult,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_4175_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_4176_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_4177_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_4178_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_4179_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).

% diff_le_diff_pow
thf(fact_4180_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% div_geq
thf(fact_4181_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N4: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M5 @ N4 )
            | ( N4 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ) ).

% div_if
thf(fact_4182_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% le_div_geq
thf(fact_4183_power__sub,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_sub
thf(fact_4184_nat__mult__power__less__eq,axiom,
    ! [B: nat,A: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ B @ N ) ) @ ( power_power_nat @ B @ M ) )
        = ( ord_less_nat @ A @ ( power_power_nat @ B @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).

% nat_mult_power_less_eq
thf(fact_4185_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_4186_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_4187_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_4188_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_4189_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_4190_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_4191_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
       => ( ( ord_less_eq_rat @ R2 @ S )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_4192_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
       => ( ( ord_less_eq_real @ R2 @ S )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_4193_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4194_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4195_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4196_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_4197_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_4198_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_4199_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_4200_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_4201_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_4202_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).

% Suc_div_eq_add3_div
thf(fact_4203_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_4204_nat__le__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( ( ord_less_eq_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ K ) ) )
     => ( ( ord_less_eq_nat @ K @ M )
       => ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_le_power_trans
thf(fact_4205_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_4206_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_4207_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( 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 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4208_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( 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 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4209_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4210_less__two__pow__divI,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ord_less_nat @ X @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% less_two_pow_divI
thf(fact_4211_less__two__pow__divD,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N )
        & ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% less_two_pow_divD
thf(fact_4212_nat__power__less__diff,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Q2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     => ( ord_less_nat @ Q2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% nat_power_less_diff
thf(fact_4213_nat__less__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ K ) ) )
     => ( ( ord_less_eq_nat @ K @ M )
       => ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_less_power_trans
thf(fact_4214_power__mod__div,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( modulo_modulo_nat @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% power_mod_div
thf(fact_4215_power2__diff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_4216_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_4217_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_4218_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_4219_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_4220_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_4221_even__mask__div__iff_H,axiom,
    ! [M: 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 ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_4222_even__mask__div__iff_H,axiom,
    ! [M: 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 ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_4223_atLeastLessThan__subset__iff,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D2 ) )
     => ( ( ord_less_eq_real @ B @ A )
        | ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4224_atLeastLessThan__subset__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C @ D2 ) )
     => ( ( ord_less_eq_num @ B @ A )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4225_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
     => ( ( ord_less_eq_nat @ B @ A )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4226_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D2 ) )
     => ( ( ord_less_eq_int @ B @ A )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4227_atLeastLessThan__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or8404916559141939852nteger @ A @ B ) @ ( set_or8404916559141939852nteger @ C @ D2 ) )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
        | ( ( ord_le3102999989581377725nteger @ C @ A )
          & ( ord_le3102999989581377725nteger @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4228_atLeastLessThan__eq__iff,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D2 )
       => ( ( ( set_or66887138388493659n_real @ A @ B )
            = ( set_or66887138388493659n_real @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4229_atLeastLessThan__eq__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( ( set_or4029947393144176647an_rat @ A @ B )
            = ( set_or4029947393144176647an_rat @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4230_atLeastLessThan__eq__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ C @ D2 )
       => ( ( ( set_or1222409239386451017an_num @ A @ B )
            = ( set_or1222409239386451017an_num @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4231_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ( set_or4665077453230672383an_nat @ A @ B )
            = ( set_or4665077453230672383an_nat @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4232_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ( set_or4662586982721622107an_int @ A @ B )
            = ( set_or4662586982721622107an_int @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4233_atLeastLessThan__eq__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ( set_or8404916559141939852nteger @ A @ B )
            = ( set_or8404916559141939852nteger @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4234_atLeastLessThan__inj_I1_J,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ( set_or66887138388493659n_real @ A @ B )
        = ( set_or66887138388493659n_real @ C @ D2 ) )
     => ( ( ord_less_real @ A @ B )
       => ( ( ord_less_real @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4235_atLeastLessThan__inj_I1_J,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = ( set_or4029947393144176647an_rat @ C @ D2 ) )
     => ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_rat @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4236_atLeastLessThan__inj_I1_J,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = ( set_or1222409239386451017an_num @ C @ D2 ) )
     => ( ( ord_less_num @ A @ B )
       => ( ( ord_less_num @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4237_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C @ D2 ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4238_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C @ D2 ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4239_atLeastLessThan__inj_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = ( set_or8404916559141939852nteger @ C @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( ord_le6747313008572928689nteger @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4240_atLeastLessThan__inj_I2_J,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ( set_or66887138388493659n_real @ A @ B )
        = ( set_or66887138388493659n_real @ C @ D2 ) )
     => ( ( ord_less_real @ A @ B )
       => ( ( ord_less_real @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4241_atLeastLessThan__inj_I2_J,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = ( set_or4029947393144176647an_rat @ C @ D2 ) )
     => ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_rat @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4242_atLeastLessThan__inj_I2_J,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = ( set_or1222409239386451017an_num @ C @ D2 ) )
     => ( ( ord_less_num @ A @ B )
       => ( ( ord_less_num @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4243_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C @ D2 ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4244_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C @ D2 ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4245_atLeastLessThan__inj_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = ( set_or8404916559141939852nteger @ C @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( ord_le6747313008572928689nteger @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4246_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
          = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).

% int_power_div_base
thf(fact_4247_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4248_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_4249_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_4250_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4251_even__mask__div__iff,axiom,
    ! [M: 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 ) ) @ M ) @ 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 @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4252_even__mask__div__iff,axiom,
    ! [M: 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 ) ) @ M ) @ 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 @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4253_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4254_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4255_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger
        @ ( zero_n356916108424825756nteger
          @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
             != zero_z3403309356797280102nteger )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4256_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_4257_cnt__cnt__eq,axiom,
    ( vEBT_VEBT_cnt
    = ( ^ [T2: vEBT_VEBT] : ( semiri5074537144036343181t_real @ ( vEBT_VEBT_cnt2 @ T2 ) ) ) ) ).

% cnt_cnt_eq
thf(fact_4258_space__cnt,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_cnt @ T ) ) ) ).

% space_cnt
thf(fact_4259_vebt__buildup__bound,axiom,
    ! [U: nat,N: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ord_less_eq_nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ U ) ) ) ).

% vebt_buildup_bound
thf(fact_4260_Tb__T__vebt__buildupi_H_H,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( vEBT_V441764108873111860ildupi @ N ) @ ( minus_minus_nat @ ( vEBT_VEBT_Tb2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi''
thf(fact_4261_T__vebt__buildupi__cnt_H,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V441764108873111860ildupi @ N ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) ) ) ).

% T_vebt_buildupi_cnt'
thf(fact_4262_count__buildup_H,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% count_buildup'
thf(fact_4263_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q2: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_4264_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q2: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_4265_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q2: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_4266_count__buildup,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).

% count_buildup
thf(fact_4267_idiff__0__right,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
      = N ) ).

% idiff_0_right
thf(fact_4268_idiff__0,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_4269_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( numeral_numeral_int @ V ) )
      = ( M
        = ( numeral_numeral_nat @ V ) ) ) ).

% int_eq_iff_numeral
thf(fact_4270_ivl__diff,axiom,
    ! [I: real,N: real,M: real] :
      ( ( ord_less_eq_real @ I @ N )
     => ( ( minus_minus_set_real @ ( set_or66887138388493659n_real @ I @ M ) @ ( set_or66887138388493659n_real @ I @ N ) )
        = ( set_or66887138388493659n_real @ N @ M ) ) ) ).

% ivl_diff
thf(fact_4271_ivl__diff,axiom,
    ! [I: num,N: num,M: num] :
      ( ( ord_less_eq_num @ I @ N )
     => ( ( minus_minus_set_num @ ( set_or1222409239386451017an_num @ I @ M ) @ ( set_or1222409239386451017an_num @ I @ N ) )
        = ( set_or1222409239386451017an_num @ N @ M ) ) ) ).

% ivl_diff
thf(fact_4272_ivl__diff,axiom,
    ! [I: nat,N: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
        = ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).

% ivl_diff
thf(fact_4273_ivl__diff,axiom,
    ! [I: int,N: int,M: int] :
      ( ( ord_less_eq_int @ I @ N )
     => ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N ) )
        = ( set_or4662586982721622107an_int @ N @ M ) ) ) ).

% ivl_diff
thf(fact_4274_ivl__diff,axiom,
    ! [I: code_integer,N: code_integer,M: code_integer] :
      ( ( ord_le3102999989581377725nteger @ I @ N )
     => ( ( minus_2355218937544613996nteger @ ( set_or8404916559141939852nteger @ I @ M ) @ ( set_or8404916559141939852nteger @ I @ N ) )
        = ( set_or8404916559141939852nteger @ N @ M ) ) ) ).

% ivl_diff
thf(fact_4275_int__dvd__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% int_dvd_int_iff
thf(fact_4276_Tb_H__cnt,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( vEBT_VEBT_Tb2 @ N ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_cnt2 @ ( vEBT_vebt_buildup @ N ) ) ) ) ).

% Tb'_cnt
thf(fact_4277_t__buildup__cnt,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8346862874174094_d_u_p @ N ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ).

% t_buildup_cnt
thf(fact_4278_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_4279_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_4280_int__distrib_I3_J,axiom,
    ! [Z1: int,Z2: int,W: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).

% int_distrib(3)
thf(fact_4281_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z2: int] :
      ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
      = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).

% int_distrib(4)
thf(fact_4282_Word_Omod__minus__cong,axiom,
    ! [B: int,B4: int,X: int,X6: int,Y: int,Y7: int,Z4: int] :
      ( ( B = B4 )
     => ( ( ( modulo_modulo_int @ X @ B4 )
          = ( modulo_modulo_int @ X6 @ B4 ) )
       => ( ( ( modulo_modulo_int @ Y @ B4 )
            = ( modulo_modulo_int @ Y7 @ B4 ) )
         => ( ( ( minus_minus_int @ X6 @ Y7 )
              = Z4 )
           => ( ( modulo_modulo_int @ ( minus_minus_int @ X @ Y ) @ B )
              = ( modulo_modulo_int @ Z4 @ B4 ) ) ) ) ) ) ).

% Word.mod_minus_cong
thf(fact_4283_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).

% signed_take_bit_diff
thf(fact_4284_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
     => ( ( dvd_dvd_int @ K @ N )
       => ( dvd_dvd_int @ K @ M ) ) ) ).

% zdvd_zdiffD
thf(fact_4285_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_4286_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_4287_int__ops_I3_J,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% int_ops(3)
thf(fact_4288_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_4289_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_4290_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% zle_int
thf(fact_4291_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_4292_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_4293_int__plus,axiom,
    ! [N: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_4294_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nonneg_int_cases
thf(fact_4295_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_4296_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_4297_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W2: int,Z7: int] :
        ? [N4: nat] :
          ( Z7
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_4298_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_4299_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_4300_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_4301_zmod__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zmod_int
thf(fact_4302_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_4303_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_4304_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_4305_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W2: int,Z7: int] :
        ? [N4: nat] :
          ( Z7
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_4306_plusinfinity,axiom,
    ! [D2: int,P2: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K2: int] :
            ( ( P2 @ X3 )
            = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z5 @ X3 )
             => ( ( P @ X3 )
                = ( P2 @ X3 ) ) )
         => ( ? [X_12: int] : ( P2 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_4307_minusinfinity,axiom,
    ! [D2: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K2: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z5 )
             => ( ( P @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_4308_less__1__helper,axiom,
    ! [N: int,M: int] :
      ( ( ord_less_eq_int @ N @ M )
     => ( ord_less_int @ ( minus_minus_int @ N @ one_one_int ) @ M ) ) ).

% less_1_helper
thf(fact_4309_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I3: int] :
            ( ( ord_less_eq_int @ K @ I3 )
           => ( ( P @ I3 )
             => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_4310_int__mod__le_H,axiom,
    ! [B: int,N: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ B @ N ) )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ B @ N ) @ ( minus_minus_int @ B @ N ) ) ) ).

% int_mod_le'
thf(fact_4311_mod__div__equality__div__eq,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ B )
      = ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_div_equality_div_eq
thf(fact_4312_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% pos_int_cases
thf(fact_4313_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( K
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_4314_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J2: int,K: nat] :
      ( ( ord_less_int @ I @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J2 ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_4315_decr__mult__lemma,axiom,
    ! [D2: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D2 ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X4: int] :
              ( ( P @ X4 )
             => ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_4316_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).

% mod_pos_geq
thf(fact_4317_int__div__sub__1,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ one_one_int @ M )
     => ( ( ( dvd_dvd_int @ M @ N )
         => ( ( divide_divide_int @ ( minus_minus_int @ N @ one_one_int ) @ M )
            = ( minus_minus_int @ ( divide_divide_int @ N @ M ) @ one_one_int ) ) )
        & ( ~ ( dvd_dvd_int @ M @ N )
         => ( ( divide_divide_int @ ( minus_minus_int @ N @ one_one_int ) @ M )
            = ( divide_divide_int @ N @ M ) ) ) ) ) ).

% int_div_sub_1
thf(fact_4318_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_diff_iff
thf(fact_4319_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_4320_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_4321_mod__sub__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( ( ord_less_int @ X @ Z )
     => ( ( ord_less_int @ Y @ Z )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( ( ord_less_eq_int @ zero_zero_int @ X )
           => ( ( ord_less_eq_int @ zero_zero_int @ Z )
             => ( ( ( ord_less_eq_int @ Y @ X )
                 => ( ( modulo_modulo_int @ ( minus_minus_int @ X @ Y ) @ Z )
                    = ( minus_minus_int @ X @ Y ) ) )
                & ( ~ ( ord_less_eq_int @ Y @ X )
                 => ( ( modulo_modulo_int @ ( minus_minus_int @ X @ Y ) @ Z )
                    = ( plus_plus_int @ ( minus_minus_int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_sub_if_z
thf(fact_4322_mod__add__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( ( ord_less_int @ X @ Z )
     => ( ( ord_less_int @ Y @ Z )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( ( ord_less_eq_int @ zero_zero_int @ X )
           => ( ( ord_less_eq_int @ zero_zero_int @ Z )
             => ( ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ Z )
                 => ( ( modulo_modulo_int @ ( plus_plus_int @ X @ Y ) @ Z )
                    = ( plus_plus_int @ X @ Y ) ) )
                & ( ~ ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ Z )
                 => ( ( modulo_modulo_int @ ( plus_plus_int @ X @ Y ) @ Z )
                    = ( minus_minus_int @ ( plus_plus_int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_add_if_z
thf(fact_4323_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    ( ( vEBT_VEBT_Tb2 @ zero_zero_nat )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb'.simps(1)
thf(fact_4324_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    ( ( vEBT_V8346862874174094_d_u_p @ zero_zero_nat )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
thf(fact_4325_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_4326_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    ( ( vEBT_VEBT_Tb2 @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb'.simps(2)
thf(fact_4327_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
thf(fact_4328_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_4329_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_4330_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.simps(3)
thf(fact_4331_VEBT__internal_OTb_H_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
         => ~ ! [N2: nat] :
                ( ( X
                  = ( suc @ ( suc @ N2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
thf(fact_4332_space__space_H,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_nat @ ( vEBT_VEBT_space @ T ) @ ( vEBT_VEBT_space2 @ T ) ) ).

% space_space'
thf(fact_4333_t__build__cnt,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ).

% t_build_cnt
thf(fact_4334_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_4335_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_4336_Tb__T__vebt__buildupi,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N ) ) @ ( minus_minus_int @ ( vEBT_VEBT_Tb @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi
thf(fact_4337_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_4338_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = zero_zero_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = one_one_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).

% mod_exhaust_less_4
thf(fact_4339_buildup__build__time,axiom,
    ! [N: nat] : ( ord_less_nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) ).

% buildup_build_time
thf(fact_4340_Tb__Tb_H,axiom,
    ( vEBT_VEBT_Tb
    = ( ^ [T2: nat] : ( semiri1314217659103216013at_int @ ( vEBT_VEBT_Tb2 @ T2 ) ) ) ) ).

% Tb_Tb'
thf(fact_4341_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M3: nat,N2: nat] :
          ( Z
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% int_diff_cases
thf(fact_4342_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% int_int_eq
thf(fact_4343_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_4344_VEBT__internal_OTb_Osimps_I1_J,axiom,
    ( ( vEBT_VEBT_Tb @ zero_zero_nat )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb.simps(1)
thf(fact_4345_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    ( ( vEBT_V8646137997579335489_i_l_d @ zero_zero_nat )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
thf(fact_4346_VEBT__internal_OTb_Osimps_I2_J,axiom,
    ( ( vEBT_VEBT_Tb @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb.simps(2)
thf(fact_4347_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
thf(fact_4348_Bolzano,axiom,
    ! [A: real,B: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A6: real,B5: real,C2: real] :
            ( ( P @ A6 @ B5 )
           => ( ( P @ B5 @ C2 )
             => ( ( ord_less_eq_real @ A6 @ B5 )
               => ( ( ord_less_eq_real @ B5 @ C2 )
                 => ( P @ A6 @ C2 ) ) ) ) )
       => ( ! [X3: real] :
              ( ( ord_less_eq_real @ A @ X3 )
             => ( ( ord_less_eq_real @ X3 @ B )
               => ? [D5: real] :
                    ( ( ord_less_real @ zero_zero_real @ D5 )
                    & ! [A6: real,B5: real] :
                        ( ( ( ord_less_eq_real @ A6 @ X3 )
                          & ( ord_less_eq_real @ X3 @ B5 )
                          & ( ord_less_real @ ( minus_minus_real @ B5 @ A6 ) @ D5 ) )
                       => ( P @ A6 @ B5 ) ) ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_4349_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.simps(3)
thf(fact_4350_VEBT__internal_OTb_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_VEBT_Tb @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
         => ~ ! [N2: nat] :
                ( ( X
                  = ( suc @ ( suc @ N2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
thf(fact_4351_Tb__T__vebt__buildupi_H,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( vEBT_V9176841429113362141ildupi @ N ) @ ( minus_minus_int @ ( vEBT_VEBT_Tb @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi'
thf(fact_4352_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) @ one_one_real ) ) ) ) ).

% space_2_pow_bound
thf(fact_4353_Tbuildupi__buildupi_H,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N ) )
      = ( vEBT_V9176841429113362141ildupi @ N ) ) ).

% Tbuildupi_buildupi'
thf(fact_4354_cnt__bound,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cnt_bound
thf(fact_4355_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_4356_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_4357_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) @ one_one_real ) ) ) ) ).

% cnt_bound'
thf(fact_4358_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != one_one_int ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y != one_one_int ) )
         => ~ ! [N2: nat] :
                ( ( X
                  = ( suc @ ( suc @ N2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
thf(fact_4359_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_4360_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_4361_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_4362_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_4363_space_H__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_nat @ ( vEBT_VEBT_space2 @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).

% space'_bound
thf(fact_4364_space__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_nat @ ( vEBT_VEBT_space @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).

% space_bound
thf(fact_4365_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    ( ( vEBT_V9176841429113362141ildupi @ zero_zero_nat )
    = one_one_int ) ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
thf(fact_4366_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    ( ( vEBT_V9176841429113362141ildupi @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
thf(fact_4367_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N2: nat] :
            ( ~ ( P @ N2 )
            & ( P @ ( suc @ N2 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_4368_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N2: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% reals_Archimedean2
thf(fact_4369_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% reals_Archimedean2
thf(fact_4370_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_4371_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_4372_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
thf(fact_4373_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpyd
thf(fact_4374_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpypredd
thf(fact_4375_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_VEBT_height @ T ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% two_powr_height_bound_deg
thf(fact_4376_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_4377_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_4378_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).

% set_n_deg_not_0
thf(fact_4379_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( some_nat @ M )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% misiz
thf(fact_4380_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_4381_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S3 ) ) ) ).

% deg_SUcn_Node
thf(fact_4382_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_4383_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_4384_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ Summary ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).

% height_compose_summary
thf(fact_4385_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_4386_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ) ).

% height_compose_child
thf(fact_4387_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_4388_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_4389_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_4390_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 )
          & ? [M3: nat] :
              ( ( ( some_nat @ M3 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_4391_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B3: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B3 )
      = ( ! [X2: complex] :
            ( ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
           => ( member_complex @ X2 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_4392_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B3 )
      = ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( member_VEBT_VEBT @ X2 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_4393_subset__code_I1_J,axiom,
    ! [Xs: list_real,B3: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B3 )
      = ( ! [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
           => ( member_real @ X2 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_4394_subset__code_I1_J,axiom,
    ! [Xs: list_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B3 )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
           => ( member_int @ X2 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_4395_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B3 )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X2 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_4396_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs4: list_complex,Xsi: list_complex,Xsi2: list_complex,A2: complex > complex > assn,A7: complex > complex > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: complex,Xi: complex] :
              ( ( member_complex @ X3 @ ( set_complex2 @ Xs4 ) )
             => ( ( member_complex @ Xi @ ( set_complex2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A7 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L4260503343685368993omplex @ A2 @ Xs @ Xsi )
            = ( vEBT_L4260503343685368993omplex @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_4397_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs4: list_complex,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: complex > vEBT_VEBT > assn,A7: complex > vEBT_VEBT > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: complex,Xi: vEBT_VEBT] :
              ( ( member_complex @ X3 @ ( set_complex2 @ Xs4 ) )
             => ( ( member_VEBT_VEBT @ Xi @ ( set_VEBT_VEBT2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A7 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L8524933119956041985T_VEBT @ A2 @ Xs @ Xsi )
            = ( vEBT_L8524933119956041985T_VEBT @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_4398_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs4: list_complex,Xsi: list_real,Xsi2: list_real,A2: complex > real > assn,A7: complex > real > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: complex,Xi: real] :
              ( ( member_complex @ X3 @ ( set_complex2 @ Xs4 ) )
             => ( ( member_real @ Xi @ ( set_real2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A7 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L2479436891206192927x_real @ A2 @ Xs @ Xsi )
            = ( vEBT_L2479436891206192927x_real @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_4399_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs4: list_complex,Xsi: list_nat,Xsi2: list_nat,A2: complex > nat > assn,A7: complex > nat > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: complex,Xi: nat] :
              ( ( member_complex @ X3 @ ( set_complex2 @ Xs4 ) )
             => ( ( member_nat @ Xi @ ( set_nat2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A7 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L137475477348087235ex_nat @ A2 @ Xs @ Xsi )
            = ( vEBT_L137475477348087235ex_nat @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_4400_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs4: list_complex,Xsi: list_int,Xsi2: list_int,A2: complex > int > assn,A7: complex > int > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: complex,Xi: int] :
              ( ( member_complex @ X3 @ ( set_complex2 @ Xs4 ) )
             => ( ( member_int @ Xi @ ( set_int2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A7 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L134985006839036959ex_int @ A2 @ Xs @ Xsi )
            = ( vEBT_L134985006839036959ex_int @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_4401_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_complex,Xsi2: list_complex,A2: vEBT_VEBT > complex > assn,A7: vEBT_VEBT > complex > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: vEBT_VEBT,Xi: complex] :
              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs4 ) )
             => ( ( member_complex @ Xi @ ( set_complex2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A7 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L2162147798726695391omplex @ A2 @ Xs @ Xsi )
            = ( vEBT_L2162147798726695391omplex @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_4402_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: vEBT_VEBT > vEBT_VEBT > assn,A7: vEBT_VEBT > vEBT_VEBT > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: vEBT_VEBT,Xi: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs4 ) )
             => ( ( member_VEBT_VEBT @ Xi @ ( set_VEBT_VEBT2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A7 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L1279224858307276611T_VEBT @ A2 @ Xs @ Xsi )
            = ( vEBT_L1279224858307276611T_VEBT @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_4403_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real,A2: vEBT_VEBT > real > assn,A7: vEBT_VEBT > real > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: vEBT_VEBT,Xi: real] :
              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs4 ) )
             => ( ( member_real @ Xi @ ( set_real2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A7 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L5781919052683127133T_real @ A2 @ Xs @ Xsi )
            = ( vEBT_L5781919052683127133T_real @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_4404_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat,A2: vEBT_VEBT > nat > assn,A7: vEBT_VEBT > nat > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: vEBT_VEBT,Xi: nat] :
              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs4 ) )
             => ( ( member_nat @ Xi @ ( set_nat2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A7 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L8296926524756676353BT_nat @ A2 @ Xs @ Xsi )
            = ( vEBT_L8296926524756676353BT_nat @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_4405_list__assn__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_int,Xsi2: list_int,A2: vEBT_VEBT > int > assn,A7: vEBT_VEBT > int > assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: vEBT_VEBT,Xi: int] :
              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs4 ) )
             => ( ( member_int @ Xi @ ( set_int2 @ Xsi2 ) )
               => ( ( A2 @ X3 @ Xi )
                  = ( A7 @ X3 @ Xi ) ) ) )
         => ( ( vEBT_L8294436054247626077BT_int @ A2 @ Xs @ Xsi )
            = ( vEBT_L8294436054247626077BT_int @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_4406_length__pos__if__in__set,axiom,
    ! [X: complex,Xs: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4407_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4408_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4409_length__pos__if__in__set,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4410_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4411_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_4412_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% delete_bound_height
thf(fact_4413_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X ) @ one_one_nat ) ) ).

% tdeletemimi'
thf(fact_4414_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% tdeletemimi
thf(fact_4415_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_4416_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_4417_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% delete_bound_height'
thf(fact_4418_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_4419_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_4420_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_4421_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 )
        & ! [Z7: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z7 )
              & ( ord_less_nat @ X @ Z7 ) )
           => ( ord_less_eq_nat @ Y @ Z7 ) ) ) ) ).

% succ_member
thf(fact_4422_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ Y @ X )
        & ! [Z7: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z7 )
              & ( ord_less_nat @ Z7 @ X ) )
           => ( ord_less_eq_nat @ Z7 @ Y ) ) ) ) ).

% pred_member
thf(fact_4423_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_corr
thf(fact_4424_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).

% pred_corr
thf(fact_4425_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_correct
thf(fact_4426_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% pred_correct
thf(fact_4427_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_4428_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_4429_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_4430_height__node,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 @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% height_node
thf(fact_4431_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_4432_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
thf(fact_4433_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% delete_bound_size_univ
thf(fact_4434_inthall,axiom,
    ! [Xs: list_VEBT_VEBTi,P: vEBT_VEBTi > $o,N: nat] :
      ( ! [X3: vEBT_VEBTi] :
          ( ( member_VEBT_VEBTi @ X3 @ ( set_VEBT_VEBTi2 @ Xs ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( P @ ( nth_VEBT_VEBTi @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4435_inthall,axiom,
    ! [Xs: list_complex,P: complex > $o,N: nat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
       => ( P @ ( nth_complex @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4436_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4437_inthall,axiom,
    ! [Xs: list_real,P: real > $o,N: nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
       => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4438_inthall,axiom,
    ! [Xs: list_o,P: $o > $o,N: nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
       => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4439_inthall,axiom,
    ! [Xs: list_nat,P: nat > $o,N: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4440_inthall,axiom,
    ! [Xs: list_int,P: int > $o,N: nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_4441_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Ma @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_4442_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( ! [Y5: product_prod_nat_nat] :
            ( X
           != ( some_P7363390416028606310at_nat @ Y5 ) ) )
      = ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_4443_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ( ! [Y5: nat] :
            ( X
           != ( some_nat @ Y5 ) ) )
      = ( X = none_nat ) ) ).

% not_Some_eq
thf(fact_4444_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ( ! [Y5: num] :
            ( X
           != ( some_num @ Y5 ) ) )
      = ( X = none_num ) ) ).

% not_Some_eq
thf(fact_4445_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
      = ( ? [Y5: product_prod_nat_nat] :
            ( X
            = ( some_P7363390416028606310at_nat @ Y5 ) ) ) ) ).

% not_None_eq
thf(fact_4446_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
      = ( ? [Y5: nat] :
            ( X
            = ( some_nat @ Y5 ) ) ) ) ).

% not_None_eq
thf(fact_4447_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
      = ( ? [Y5: num] :
            ( X
            = ( some_num @ Y5 ) ) ) ) ).

% not_None_eq
thf(fact_4448_less__eq__option__None__code,axiom,
    ! [X: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X ) ).

% less_eq_option_None_code
thf(fact_4449_less__eq__option__None__code,axiom,
    ! [X: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X ) ).

% less_eq_option_None_code
thf(fact_4450_less__option__None,axiom,
    ! [X: option_nat] :
      ~ ( ord_less_option_nat @ X @ none_nat ) ).

% less_option_None
thf(fact_4451_less__option__None,axiom,
    ! [X: option_num] :
      ~ ( ord_less_option_num @ X @ none_num ) ).

% less_option_None
thf(fact_4452_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% delete_bound_size_univ'
thf(fact_4453_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: vEBT_VEBT] :
      ( ( U
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Deg ) )
     => ( ( vEBT_invar_vebt @ T @ Deg )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_height @ T ) ) @ ( plus_plus_real @ one_one_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% height_double_log_univ_size
thf(fact_4454_less__eq__option__Some__None,axiom,
    ! [X: nat] :
      ~ ( ord_le5914376470875661696on_nat @ ( some_nat @ X ) @ none_nat ) ).

% less_eq_option_Some_None
thf(fact_4455_less__eq__option__Some__None,axiom,
    ! [X: num] :
      ~ ( ord_le6622620407824499402on_num @ ( some_num @ X ) @ none_num ) ).

% less_eq_option_Some_None
thf(fact_4456_less__option__None__Some__code,axiom,
    ! [X: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X ) ) ).

% less_option_None_Some_code
thf(fact_4457_less__option__None__Some__code,axiom,
    ! [X: num] : ( ord_less_option_num @ none_num @ ( some_num @ X ) ) ).

% less_option_None_Some_code
thf(fact_4458_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 ) )
       => ( ! [A6: product_prod_nat_nat,B5: product_prod_nat_nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A6 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_4459_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 ) )
       => ( ! [A6: product_prod_nat_nat,B5: nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A6 ) )
             => ( ( Y
                  = ( some_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_4460_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 ) )
       => ( ! [A6: product_prod_nat_nat,B5: num] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A6 ) )
             => ( ( Y
                  = ( some_num @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_4461_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 ) )
       => ( ! [A6: nat,B5: product_prod_nat_nat] :
              ( ( X
                = ( some_nat @ A6 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_4462_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 ) )
       => ( ! [A6: nat,B5: nat] :
              ( ( X
                = ( some_nat @ A6 ) )
             => ( ( Y
                  = ( some_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_4463_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 ) )
       => ( ! [A6: nat,B5: num] :
              ( ( X
                = ( some_nat @ A6 ) )
             => ( ( Y
                  = ( some_num @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_4464_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 ) )
       => ( ! [A6: num,B5: product_prod_nat_nat] :
              ( ( X
                = ( some_num @ A6 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_4465_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 ) )
       => ( ! [A6: num,B5: nat] :
              ( ( X
                = ( some_num @ A6 ) )
             => ( ( Y
                  = ( some_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_4466_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 ) )
       => ( ! [A6: num,B5: num] :
              ( ( X
                = ( some_num @ A6 ) )
             => ( ( Y
                  = ( some_num @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_4467_split__option__all,axiom,
    ( ( ^ [P5: option4927543243414619207at_nat > $o] :
        ! [X7: option4927543243414619207at_nat] : ( P5 @ X7 ) )
    = ( ^ [P6: option4927543243414619207at_nat > $o] :
          ( ( P6 @ none_P5556105721700978146at_nat )
          & ! [X2: product_prod_nat_nat] : ( P6 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_4468_split__option__all,axiom,
    ( ( ^ [P5: option_nat > $o] :
        ! [X7: option_nat] : ( P5 @ X7 ) )
    = ( ^ [P6: option_nat > $o] :
          ( ( P6 @ none_nat )
          & ! [X2: nat] : ( P6 @ ( some_nat @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_4469_split__option__all,axiom,
    ( ( ^ [P5: option_num > $o] :
        ! [X7: option_num] : ( P5 @ X7 ) )
    = ( ^ [P6: option_num > $o] :
          ( ( P6 @ none_num )
          & ! [X2: num] : ( P6 @ ( some_num @ X2 ) ) ) ) ) ).

% split_option_all
thf(fact_4470_split__option__ex,axiom,
    ( ( ^ [P5: option4927543243414619207at_nat > $o] :
        ? [X7: option4927543243414619207at_nat] : ( P5 @ X7 ) )
    = ( ^ [P6: option4927543243414619207at_nat > $o] :
          ( ( P6 @ none_P5556105721700978146at_nat )
          | ? [X2: product_prod_nat_nat] : ( P6 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_4471_split__option__ex,axiom,
    ( ( ^ [P5: option_nat > $o] :
        ? [X7: option_nat] : ( P5 @ X7 ) )
    = ( ^ [P6: option_nat > $o] :
          ( ( P6 @ none_nat )
          | ? [X2: nat] : ( P6 @ ( some_nat @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_4472_split__option__ex,axiom,
    ( ( ^ [P5: option_num > $o] :
        ? [X7: option_num] : ( P5 @ X7 ) )
    = ( ^ [P6: option_num > $o] :
          ( ( P6 @ none_num )
          | ? [X2: num] : ( P6 @ ( some_num @ X2 ) ) ) ) ) ).

% split_option_ex
thf(fact_4473_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] :
            ( Y
           != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_4474_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] :
            ( Y
           != ( some_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_4475_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] :
            ( Y
           != ( some_num @ X23 ) ) ) ).

% option.exhaust
thf(fact_4476_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X22 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_4477_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option
        = ( some_nat @ X22 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_4478_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option
        = ( some_num @ X22 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_4479_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_4480_option_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( none_nat
     != ( some_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_4481_option_Odistinct_I1_J,axiom,
    ! [X22: num] :
      ( none_num
     != ( some_num @ X22 ) ) ).

% option.distinct(1)
thf(fact_4482_less__eq__option__None,axiom,
    ! [X: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X ) ).

% less_eq_option_None
thf(fact_4483_less__eq__option__None,axiom,
    ! [X: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X ) ).

% less_eq_option_None
thf(fact_4484_less__eq__option__None__is__None,axiom,
    ! [X: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ X @ none_nat )
     => ( X = none_nat ) ) ).

% less_eq_option_None_is_None
thf(fact_4485_less__eq__option__None__is__None,axiom,
    ! [X: option_num] :
      ( ( ord_le6622620407824499402on_num @ X @ none_num )
     => ( X = none_num ) ) ).

% less_eq_option_None_is_None
thf(fact_4486_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_VEBT_VEBT,Z3: list_VEBT_VEBT] : Y6 = Z3 )
    = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
             => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
                = ( nth_VEBT_VEBT @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_4487_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_VEBT_VEBTi,Z3: list_VEBT_VEBTi] : Y6 = Z3 )
    = ( ^ [Xs2: list_VEBT_VEBTi,Ys3: list_VEBT_VEBTi] :
          ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
            = ( size_s7982070591426661849_VEBTi @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
             => ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
                = ( nth_VEBT_VEBTi @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_4488_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_real,Z3: list_real] : Y6 = Z3 )
    = ( ^ [Xs2: list_real,Ys3: list_real] :
          ( ( ( size_size_list_real @ Xs2 )
            = ( size_size_list_real @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
             => ( ( nth_real @ Xs2 @ I2 )
                = ( nth_real @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_4489_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_o,Z3: list_o] : Y6 = Z3 )
    = ( ^ [Xs2: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs2 )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
             => ( ( nth_o @ Xs2 @ I2 )
                = ( nth_o @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_4490_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_nat,Z3: list_nat] : Y6 = Z3 )
    = ( ^ [Xs2: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I2 )
                = ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_4491_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_int,Z3: list_int] : Y6 = Z3 )
    = ( ^ [Xs2: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs2 )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
             => ( ( nth_int @ Xs2 @ I2 )
                = ( nth_int @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_4492_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X8: vEBT_VEBT] : ( P @ I2 @ X8 ) ) )
      = ( ? [Xs2: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_4493_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBTi > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X8: vEBT_VEBTi] : ( P @ I2 @ X8 ) ) )
      = ( ? [Xs2: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_VEBT_VEBTi @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_4494_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > real > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X8: real] : ( P @ I2 @ X8 ) ) )
      = ( ? [Xs2: list_real] :
            ( ( ( size_size_list_real @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_real @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_4495_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X8: $o] : ( P @ I2 @ X8 ) ) )
      = ( ? [Xs2: list_o] :
            ( ( ( size_size_list_o @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_o @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_4496_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X8: nat] : ( P @ I2 @ X8 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_4497_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X8: int] : ( P @ I2 @ X8 ) ) )
      = ( ? [Xs2: list_int] :
            ( ( ( size_size_list_int @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_int @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_4498_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( ( nth_VEBT_VEBT @ Xs @ I3 )
              = ( nth_VEBT_VEBT @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_4499_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ( size_s7982070591426661849_VEBTi @ Xs )
        = ( size_s7982070591426661849_VEBTi @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
           => ( ( nth_VEBT_VEBTi @ Xs @ I3 )
              = ( nth_VEBT_VEBTi @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_4500_nth__equalityI,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( ( size_size_list_real @ Xs )
        = ( size_size_list_real @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
           => ( ( nth_real @ Xs @ I3 )
              = ( nth_real @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_4501_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
           => ( ( nth_o @ Xs @ I3 )
              = ( nth_o @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_4502_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I3 )
              = ( nth_nat @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_4503_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
           => ( ( nth_int @ Xs @ I3 )
              = ( nth_int @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_4504_obtain__list__from__elements,axiom,
    ! [N: nat,P: vEBT_VEBT > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ? [Li: vEBT_VEBT] : ( P @ Li @ I3 ) )
     => ~ ! [L4: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ L4 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P @ ( nth_VEBT_VEBT @ L4 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_4505_obtain__list__from__elements,axiom,
    ! [N: nat,P: vEBT_VEBTi > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ? [Li: vEBT_VEBTi] : ( P @ Li @ I3 ) )
     => ~ ! [L4: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ L4 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P @ ( nth_VEBT_VEBTi @ L4 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_4506_obtain__list__from__elements,axiom,
    ! [N: nat,P: real > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ? [Li: real] : ( P @ Li @ I3 ) )
     => ~ ! [L4: list_real] :
            ( ( ( size_size_list_real @ L4 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P @ ( nth_real @ L4 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_4507_obtain__list__from__elements,axiom,
    ! [N: nat,P: $o > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ? [Li: $o] : ( P @ Li @ I3 ) )
     => ~ ! [L4: list_o] :
            ( ( ( size_size_list_o @ L4 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P @ ( nth_o @ L4 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_4508_obtain__list__from__elements,axiom,
    ! [N: nat,P: nat > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ? [Li: nat] : ( P @ Li @ I3 ) )
     => ~ ! [L4: list_nat] :
            ( ( ( size_size_list_nat @ L4 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P @ ( nth_nat @ L4 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_4509_obtain__list__from__elements,axiom,
    ! [N: nat,P: int > nat > $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ? [Li: int] : ( P @ Li @ I3 ) )
     => ~ ! [L4: list_int] :
            ( ( ( size_size_list_int @ L4 )
              = N )
           => ~ ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ N )
                 => ( P @ ( nth_int @ L4 @ I4 ) @ I4 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_4510_less__option__None__Some,axiom,
    ! [X: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X ) ) ).

% less_option_None_Some
thf(fact_4511_less__option__None__Some,axiom,
    ! [X: num] : ( ord_less_option_num @ none_num @ ( some_num @ X ) ) ).

% less_option_None_Some
thf(fact_4512_less__option__None__is__Some,axiom,
    ! [X: option_nat] :
      ( ( ord_less_option_nat @ none_nat @ X )
     => ? [Z6: nat] :
          ( X
          = ( some_nat @ Z6 ) ) ) ).

% less_option_None_is_Some
thf(fact_4513_less__option__None__is__Some,axiom,
    ! [X: option_num] :
      ( ( ord_less_option_num @ none_num @ X )
     => ? [Z6: num] :
          ( X
          = ( some_num @ Z6 ) ) ) ).

% less_option_None_is_Some
thf(fact_4514_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ! [X2: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
           => ( P @ ( nth_VEBT_VEBTi @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_4515_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_4516_all__set__conv__all__nth,axiom,
    ! [Xs: list_real,P: real > $o] :
      ( ( ! [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
           => ( P @ ( nth_real @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_4517_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: $o > $o] :
      ( ( ! [X2: $o] :
            ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
           => ( P @ ( nth_o @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_4518_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_4519_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
           => ( P @ X2 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
           => ( P @ ( nth_int @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_4520_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBTi,P: vEBT_VEBTi > $o,X: vEBT_VEBTi] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
         => ( P @ ( nth_VEBT_VEBTi @ Xs @ I3 ) ) )
     => ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4521_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: complex > $o,X: complex] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
         => ( P @ ( nth_complex @ Xs @ I3 ) ) )
     => ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4522_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4523_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: real > $o,X: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
         => ( P @ ( nth_real @ Xs @ I3 ) ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4524_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: $o > $o,X: $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
         => ( P @ ( nth_o @ Xs @ I3 ) ) )
     => ( ( member_o @ X @ ( set_o2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4525_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I3 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4526_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: int > $o,X: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
         => ( P @ ( nth_int @ Xs @ I3 ) ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_4527_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
            & ( ( nth_VEBT_VEBTi @ Xs @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4528_in__set__conv__nth,axiom,
    ! [X: complex,Xs: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs ) )
            & ( ( nth_complex @ Xs @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4529_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ( nth_VEBT_VEBT @ Xs @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4530_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
            & ( ( nth_real @ Xs @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4531_in__set__conv__nth,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
            & ( ( nth_o @ Xs @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4532_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4533_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_4534_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ! [X3: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X3 @ ( set_VEBT_VEBTi2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_VEBT_VEBTi @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_4535_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_4536_list__ball__nth,axiom,
    ! [N: nat,Xs: list_real,P: real > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_4537_list__ball__nth,axiom,
    ! [N: nat,Xs: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_4538_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_4539_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: int > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_4540_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( member_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Xs @ N ) @ ( set_VEBT_VEBTi2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4541_nth__mem,axiom,
    ! [N: nat,Xs: list_complex] :
      ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
     => ( member_complex @ ( nth_complex @ Xs @ N ) @ ( set_complex2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4542_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4543_nth__mem,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
     => ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4544_nth__mem,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4545_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4546_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).

% nth_mem
thf(fact_4547_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ! [X2: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ L ) )
           => ( P @ X2 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ L ) )
           => ( P @ ( nth_VEBT_VEBTi @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_4548_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ L ) )
           => ( P @ X2 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ L ) )
           => ( P @ ( nth_VEBT_VEBT @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_4549_all__set__conv__nth,axiom,
    ! [L: list_real,P: real > $o] :
      ( ( ! [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ L ) )
           => ( P @ X2 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ L ) )
           => ( P @ ( nth_real @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_4550_all__set__conv__nth,axiom,
    ! [L: list_o,P: $o > $o] :
      ( ( ! [X2: $o] :
            ( ( member_o @ X2 @ ( set_o2 @ L ) )
           => ( P @ X2 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ L ) )
           => ( P @ ( nth_o @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_4551_all__set__conv__nth,axiom,
    ! [L: list_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ L ) )
           => ( P @ X2 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L ) )
           => ( P @ ( nth_nat @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_4552_all__set__conv__nth,axiom,
    ! [L: list_int,P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ L ) )
           => ( P @ X2 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ L ) )
           => ( P @ ( nth_int @ L @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_4553_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: vEBT_VEBT > vEBT_VEBT > assn,A7: vEBT_VEBT > vEBT_VEBT > assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I5 = I6 )
     => ( ( A2 = A7 )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
         => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
              = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
           => ( ! [I3: nat] :
                  ( ( member_nat @ I3 @ I5 )
                 => ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
                   => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                        = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBT @ Xs @ I3 )
                          = ( nth_VEBT_VEBT @ Xs4 @ I3 ) )
                        & ( ( nth_VEBT_VEBT @ Xsi @ I3 )
                          = ( nth_VEBT_VEBT @ Xsi2 @ I3 ) ) ) ) ) )
             => ( ( vEBT_L3204528365124325536T_VEBT @ I5 @ A2 @ Xs @ Xsi )
                = ( vEBT_L3204528365124325536T_VEBT @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_4554_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: vEBT_VEBTi > vEBT_VEBT > assn,A7: vEBT_VEBTi > vEBT_VEBT > assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I5 = I6 )
     => ( ( A2 = A7 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
         => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
              = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
           => ( ! [I3: nat] :
                  ( ( member_nat @ I3 @ I5 )
                 => ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                        = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs @ I3 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I3 ) )
                        & ( ( nth_VEBT_VEBT @ Xsi @ I3 )
                          = ( nth_VEBT_VEBT @ Xsi2 @ I3 ) ) ) ) ) )
             => ( ( vEBT_L2497118539674116125T_VEBT @ I5 @ A2 @ Xs @ Xsi )
                = ( vEBT_L2497118539674116125T_VEBT @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_4555_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: vEBT_VEBTi > vEBT_VEBTi > assn,A7: vEBT_VEBTi > vEBT_VEBTi > assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi] :
      ( ( I5 = I6 )
     => ( ( A2 = A7 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
         => ( ( ( size_s7982070591426661849_VEBTi @ Xsi )
              = ( size_s7982070591426661849_VEBTi @ Xsi2 ) )
           => ( ! [I3: nat] :
                  ( ( member_nat @ I3 @ I5 )
                 => ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                        = ( size_s7982070591426661849_VEBTi @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs @ I3 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I3 ) )
                        & ( ( nth_VEBT_VEBTi @ Xsi @ I3 )
                          = ( nth_VEBT_VEBTi @ Xsi2 @ I3 ) ) ) ) ) )
             => ( ( vEBT_L886525131989349516_VEBTi @ I5 @ A2 @ Xs @ Xsi )
                = ( vEBT_L886525131989349516_VEBTi @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_4556_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: vEBT_VEBT > real > assn,A7: vEBT_VEBT > real > assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real] :
      ( ( I5 = I6 )
     => ( ( A2 = A7 )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
         => ( ( ( size_size_list_real @ Xsi )
              = ( size_size_list_real @ Xsi2 ) )
           => ( ! [I3: nat] :
                  ( ( member_nat @ I3 @ I5 )
                 => ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
                   => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                        = ( size_size_list_real @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBT @ Xs @ I3 )
                          = ( nth_VEBT_VEBT @ Xs4 @ I3 ) )
                        & ( ( nth_real @ Xsi @ I3 )
                          = ( nth_real @ Xsi2 @ I3 ) ) ) ) ) )
             => ( ( vEBT_L4281036506115550016T_real @ I5 @ A2 @ Xs @ Xsi )
                = ( vEBT_L4281036506115550016T_real @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_4557_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: vEBT_VEBTi > real > assn,A7: vEBT_VEBTi > real > assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_real,Xsi2: list_real] :
      ( ( I5 = I6 )
     => ( ( A2 = A7 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
         => ( ( ( size_size_list_real @ Xsi )
              = ( size_size_list_real @ Xsi2 ) )
           => ( ! [I3: nat] :
                  ( ( member_nat @ I3 @ I5 )
                 => ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                        = ( size_size_list_real @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs @ I3 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I3 ) )
                        & ( ( nth_real @ Xsi @ I3 )
                          = ( nth_real @ Xsi2 @ I3 ) ) ) ) ) )
             => ( ( vEBT_L7728200936804140803i_real @ I5 @ A2 @ Xs @ Xsi )
                = ( vEBT_L7728200936804140803i_real @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_4558_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: vEBT_VEBT > $o > assn,A7: vEBT_VEBT > $o > assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_o,Xsi2: list_o] :
      ( ( I5 = I6 )
     => ( ( A2 = A7 )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
         => ( ( ( size_size_list_o @ Xsi )
              = ( size_size_list_o @ Xsi2 ) )
           => ( ! [I3: nat] :
                  ( ( member_nat @ I3 @ I5 )
                 => ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
                   => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                        = ( size_size_list_o @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBT @ Xs @ I3 )
                          = ( nth_VEBT_VEBT @ Xs4 @ I3 ) )
                        & ( ( nth_o @ Xsi @ I3 )
                          = ( nth_o @ Xsi2 @ I3 ) ) ) ) ) )
             => ( ( vEBT_L7058566406413635588VEBT_o @ I5 @ A2 @ Xs @ Xsi )
                = ( vEBT_L7058566406413635588VEBT_o @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_4559_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: vEBT_VEBTi > $o > assn,A7: vEBT_VEBTi > $o > assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_o,Xsi2: list_o] :
      ( ( I5 = I6 )
     => ( ( A2 = A7 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
         => ( ( ( size_size_list_o @ Xsi )
              = ( size_size_list_o @ Xsi2 ) )
           => ( ! [I3: nat] :
                  ( ( member_nat @ I3 @ I5 )
                 => ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                        = ( size_size_list_o @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs @ I3 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I3 ) )
                        & ( ( nth_o @ Xsi @ I3 )
                          = ( nth_o @ Xsi2 @ I3 ) ) ) ) ) )
             => ( ( vEBT_L3328983362619735041EBTi_o @ I5 @ A2 @ Xs @ Xsi )
                = ( vEBT_L3328983362619735041EBTi_o @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_4560_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: vEBT_VEBT > nat > assn,A7: vEBT_VEBT > nat > assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I5 = I6 )
     => ( ( A2 = A7 )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
         => ( ( ( size_size_list_nat @ Xsi )
              = ( size_size_list_nat @ Xsi2 ) )
           => ( ! [I3: nat] :
                  ( ( member_nat @ I3 @ I5 )
                 => ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
                   => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                        = ( size_size_list_nat @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBT @ Xs @ I3 )
                          = ( nth_VEBT_VEBT @ Xs4 @ I3 ) )
                        & ( ( nth_nat @ Xsi @ I3 )
                          = ( nth_nat @ Xsi2 @ I3 ) ) ) ) ) )
             => ( ( vEBT_L8650695023172932196BT_nat @ I5 @ A2 @ Xs @ Xsi )
                = ( vEBT_L8650695023172932196BT_nat @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_4561_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: vEBT_VEBTi > nat > assn,A7: vEBT_VEBTi > nat > assn,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I5 = I6 )
     => ( ( A2 = A7 )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
         => ( ( ( size_size_list_nat @ Xsi )
              = ( size_size_list_nat @ Xsi2 ) )
           => ( ! [I3: nat] :
                  ( ( member_nat @ I3 @ I5 )
                 => ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                   => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                        = ( size_size_list_nat @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBTi @ Xs @ I3 )
                          = ( nth_VEBT_VEBTi @ Xs4 @ I3 ) )
                        & ( ( nth_nat @ Xsi @ I3 )
                          = ( nth_nat @ Xsi2 @ I3 ) ) ) ) ) )
             => ( ( vEBT_L2809031099982602151Ti_nat @ I5 @ A2 @ Xs @ Xsi )
                = ( vEBT_L2809031099982602151Ti_nat @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_4562_listI__assn__weak__cong,axiom,
    ! [I5: set_nat,I6: set_nat,A2: vEBT_VEBT > int > assn,A7: vEBT_VEBT > int > assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_int,Xsi2: list_int] :
      ( ( I5 = I6 )
     => ( ( A2 = A7 )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
         => ( ( ( size_size_list_int @ Xsi )
              = ( size_size_list_int @ Xsi2 ) )
           => ( ! [I3: nat] :
                  ( ( member_nat @ I3 @ I5 )
                 => ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
                   => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                        = ( size_size_list_int @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBT @ Xs @ I3 )
                          = ( nth_VEBT_VEBT @ Xs4 @ I3 ) )
                        & ( ( nth_int @ Xsi @ I3 )
                          = ( nth_int @ Xsi2 @ I3 ) ) ) ) ) )
             => ( ( vEBT_L8648204552663881920BT_int @ I5 @ A2 @ Xs @ Xsi )
                = ( vEBT_L8648204552663881920BT_int @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_4563_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: vEBT_VEBT > vEBT_VEBT > assn,A7: vEBT_VEBT > vEBT_VEBT > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
          = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
            = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
                 => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                      = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBT @ Xs @ I3 )
                        = ( nth_VEBT_VEBT @ Xs4 @ I3 ) )
                      & ( ( nth_VEBT_VEBT @ Xsi @ I3 )
                        = ( nth_VEBT_VEBT @ Xsi2 @ I3 ) )
                      & ( ( A2 @ ( nth_VEBT_VEBT @ Xs @ I3 ) @ ( nth_VEBT_VEBT @ Xsi @ I3 ) )
                        = ( A7 @ ( nth_VEBT_VEBT @ Xs4 @ I3 ) @ ( nth_VEBT_VEBT @ Xsi2 @ I3 ) ) ) ) ) ) )
           => ( ( vEBT_L3204528365124325536T_VEBT @ I5 @ A2 @ Xs @ Xsi )
              = ( vEBT_L3204528365124325536T_VEBT @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_4564_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A2: vEBT_VEBTi > vEBT_VEBT > assn,A7: vEBT_VEBTi > vEBT_VEBT > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
          = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
            = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                      = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs @ I3 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I3 ) )
                      & ( ( nth_VEBT_VEBT @ Xsi @ I3 )
                        = ( nth_VEBT_VEBT @ Xsi2 @ I3 ) )
                      & ( ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I3 ) @ ( nth_VEBT_VEBT @ Xsi @ I3 ) )
                        = ( A7 @ ( nth_VEBT_VEBTi @ Xs4 @ I3 ) @ ( nth_VEBT_VEBT @ Xsi2 @ I3 ) ) ) ) ) ) )
           => ( ( vEBT_L2497118539674116125T_VEBT @ I5 @ A2 @ Xs @ Xsi )
              = ( vEBT_L2497118539674116125T_VEBT @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_4565_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi,A2: vEBT_VEBTi > vEBT_VEBTi > assn,A7: vEBT_VEBTi > vEBT_VEBTi > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
          = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xsi )
            = ( size_s7982070591426661849_VEBTi @ Xsi2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                      = ( size_s7982070591426661849_VEBTi @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs @ I3 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I3 ) )
                      & ( ( nth_VEBT_VEBTi @ Xsi @ I3 )
                        = ( nth_VEBT_VEBTi @ Xsi2 @ I3 ) )
                      & ( ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I3 ) @ ( nth_VEBT_VEBTi @ Xsi @ I3 ) )
                        = ( A7 @ ( nth_VEBT_VEBTi @ Xs4 @ I3 ) @ ( nth_VEBT_VEBTi @ Xsi2 @ I3 ) ) ) ) ) ) )
           => ( ( vEBT_L886525131989349516_VEBTi @ I5 @ A2 @ Xs @ Xsi )
              = ( vEBT_L886525131989349516_VEBTi @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_4566_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real,A2: vEBT_VEBT > real > assn,A7: vEBT_VEBT > real > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
          = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
       => ( ( ( size_size_list_real @ Xsi )
            = ( size_size_list_real @ Xsi2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
                 => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                      = ( size_size_list_real @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBT @ Xs @ I3 )
                        = ( nth_VEBT_VEBT @ Xs4 @ I3 ) )
                      & ( ( nth_real @ Xsi @ I3 )
                        = ( nth_real @ Xsi2 @ I3 ) )
                      & ( ( A2 @ ( nth_VEBT_VEBT @ Xs @ I3 ) @ ( nth_real @ Xsi @ I3 ) )
                        = ( A7 @ ( nth_VEBT_VEBT @ Xs4 @ I3 ) @ ( nth_real @ Xsi2 @ I3 ) ) ) ) ) ) )
           => ( ( vEBT_L4281036506115550016T_real @ I5 @ A2 @ Xs @ Xsi )
              = ( vEBT_L4281036506115550016T_real @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_4567_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_real,Xsi2: list_real,A2: vEBT_VEBTi > real > assn,A7: vEBT_VEBTi > real > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
          = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
       => ( ( ( size_size_list_real @ Xsi )
            = ( size_size_list_real @ Xsi2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                      = ( size_size_list_real @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs @ I3 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I3 ) )
                      & ( ( nth_real @ Xsi @ I3 )
                        = ( nth_real @ Xsi2 @ I3 ) )
                      & ( ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I3 ) @ ( nth_real @ Xsi @ I3 ) )
                        = ( A7 @ ( nth_VEBT_VEBTi @ Xs4 @ I3 ) @ ( nth_real @ Xsi2 @ I3 ) ) ) ) ) ) )
           => ( ( vEBT_L7728200936804140803i_real @ I5 @ A2 @ Xs @ Xsi )
              = ( vEBT_L7728200936804140803i_real @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_4568_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_o,Xsi2: list_o,A2: vEBT_VEBT > $o > assn,A7: vEBT_VEBT > $o > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
          = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
       => ( ( ( size_size_list_o @ Xsi )
            = ( size_size_list_o @ Xsi2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
                 => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                      = ( size_size_list_o @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBT @ Xs @ I3 )
                        = ( nth_VEBT_VEBT @ Xs4 @ I3 ) )
                      & ( ( nth_o @ Xsi @ I3 )
                        = ( nth_o @ Xsi2 @ I3 ) )
                      & ( ( A2 @ ( nth_VEBT_VEBT @ Xs @ I3 ) @ ( nth_o @ Xsi @ I3 ) )
                        = ( A7 @ ( nth_VEBT_VEBT @ Xs4 @ I3 ) @ ( nth_o @ Xsi2 @ I3 ) ) ) ) ) ) )
           => ( ( vEBT_L7058566406413635588VEBT_o @ I5 @ A2 @ Xs @ Xsi )
              = ( vEBT_L7058566406413635588VEBT_o @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_4569_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_o,Xsi2: list_o,A2: vEBT_VEBTi > $o > assn,A7: vEBT_VEBTi > $o > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
          = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
       => ( ( ( size_size_list_o @ Xsi )
            = ( size_size_list_o @ Xsi2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                      = ( size_size_list_o @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs @ I3 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I3 ) )
                      & ( ( nth_o @ Xsi @ I3 )
                        = ( nth_o @ Xsi2 @ I3 ) )
                      & ( ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I3 ) @ ( nth_o @ Xsi @ I3 ) )
                        = ( A7 @ ( nth_VEBT_VEBTi @ Xs4 @ I3 ) @ ( nth_o @ Xsi2 @ I3 ) ) ) ) ) ) )
           => ( ( vEBT_L3328983362619735041EBTi_o @ I5 @ A2 @ Xs @ Xsi )
              = ( vEBT_L3328983362619735041EBTi_o @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_4570_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat,A2: vEBT_VEBT > nat > assn,A7: vEBT_VEBT > nat > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
          = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
       => ( ( ( size_size_list_nat @ Xsi )
            = ( size_size_list_nat @ Xsi2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
                 => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                      = ( size_size_list_nat @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBT @ Xs @ I3 )
                        = ( nth_VEBT_VEBT @ Xs4 @ I3 ) )
                      & ( ( nth_nat @ Xsi @ I3 )
                        = ( nth_nat @ Xsi2 @ I3 ) )
                      & ( ( A2 @ ( nth_VEBT_VEBT @ Xs @ I3 ) @ ( nth_nat @ Xsi @ I3 ) )
                        = ( A7 @ ( nth_VEBT_VEBT @ Xs4 @ I3 ) @ ( nth_nat @ Xsi2 @ I3 ) ) ) ) ) ) )
           => ( ( vEBT_L8650695023172932196BT_nat @ I5 @ A2 @ Xs @ Xsi )
              = ( vEBT_L8650695023172932196BT_nat @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_4571_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBTi,Xs4: list_VEBT_VEBTi,Xsi: list_nat,Xsi2: list_nat,A2: vEBT_VEBTi > nat > assn,A7: vEBT_VEBTi > nat > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
          = ( size_s7982070591426661849_VEBTi @ Xs4 ) )
       => ( ( ( size_size_list_nat @ Xsi )
            = ( size_size_list_nat @ Xsi2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ( ord_less_nat @ I3 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
                 => ( ( ( size_s7982070591426661849_VEBTi @ Xs )
                      = ( size_size_list_nat @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBTi @ Xs @ I3 )
                        = ( nth_VEBT_VEBTi @ Xs4 @ I3 ) )
                      & ( ( nth_nat @ Xsi @ I3 )
                        = ( nth_nat @ Xsi2 @ I3 ) )
                      & ( ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I3 ) @ ( nth_nat @ Xsi @ I3 ) )
                        = ( A7 @ ( nth_VEBT_VEBTi @ Xs4 @ I3 ) @ ( nth_nat @ Xsi2 @ I3 ) ) ) ) ) ) )
           => ( ( vEBT_L2809031099982602151Ti_nat @ I5 @ A2 @ Xs @ Xsi )
              = ( vEBT_L2809031099982602151Ti_nat @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_4572_listI__assn__cong,axiom,
    ! [I5: set_nat,I6: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_int,Xsi2: list_int,A2: vEBT_VEBT > int > assn,A7: vEBT_VEBT > int > assn] :
      ( ( I5 = I6 )
     => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
          = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
       => ( ( ( size_size_list_int @ Xsi )
            = ( size_size_list_int @ Xsi2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
                 => ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                      = ( size_size_list_int @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBT @ Xs @ I3 )
                        = ( nth_VEBT_VEBT @ Xs4 @ I3 ) )
                      & ( ( nth_int @ Xsi @ I3 )
                        = ( nth_int @ Xsi2 @ I3 ) )
                      & ( ( A2 @ ( nth_VEBT_VEBT @ Xs @ I3 ) @ ( nth_int @ Xsi @ I3 ) )
                        = ( A7 @ ( nth_VEBT_VEBT @ Xs4 @ I3 ) @ ( nth_int @ Xsi2 @ I3 ) ) ) ) ) ) )
           => ( ( vEBT_L8648204552663881920BT_int @ I5 @ A2 @ Xs @ Xsi )
              = ( vEBT_L8648204552663881920BT_int @ I6 @ A7 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_4573_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_4574_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_4575_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_4576_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_4577_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_4578_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_4579_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
thf(fact_4580_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_4581_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_4582_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_4583_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_4584_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_4585_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_4586_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_4587_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_le
thf(fact_4588_not__Some__eq2,axiom,
    ! [V: option936205604648967762et_nat] :
      ( ( ! [X2: heap_e7401611519738050253t_unit,Y5: set_nat] :
            ( V
           != ( some_P624177172695371229et_nat @ ( produc7507926704131184380et_nat @ X2 @ Y5 ) ) ) )
      = ( V = none_P533106815845188193et_nat ) ) ).

% not_Some_eq2
thf(fact_4589_not__Some__eq2,axiom,
    ! [V: option2661157926820139483um_num] :
      ( ( ! [X2: num,Y5: num] :
            ( V
           != ( some_P6201964756284913402um_num @ ( product_Pair_num_num @ X2 @ Y5 ) ) ) )
      = ( V = none_P4394680061957285238um_num ) ) ).

% not_Some_eq2
thf(fact_4590_not__Some__eq2,axiom,
    ! [V: option642762832853965969at_num] :
      ( ( ! [X2: nat,Y5: num] :
            ( V
           != ( some_P8071634352977444016at_num @ ( product_Pair_nat_num @ X2 @ Y5 ) ) ) )
      = ( V = none_P6264349658649815852at_num ) ) ).

% not_Some_eq2
thf(fact_4591_not__Some__eq2,axiom,
    ! [V: option4624381673175914239nt_int] :
      ( ( ! [X2: int,Y5: int] :
            ( V
           != ( some_P4184893108420464158nt_int @ ( product_Pair_int_int @ X2 @ Y5 ) ) ) )
      = ( V = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
thf(fact_4592_not__Some__eq2,axiom,
    ! [V: option4927543243414619207at_nat] :
      ( ( ! [X2: nat,Y5: nat] :
            ( V
           != ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ Y5 ) ) ) )
      = ( V = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
thf(fact_4593_log__one,axiom,
    ! [A: real] :
      ( ( log @ A @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_4594_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_4595_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_4596_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_4597_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_4598_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_4599_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_4600_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_4601_less__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).

% less_log_of_power
thf(fact_4602_log__of__power__eq,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( semiri5074537144036343181t_real @ N )
          = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_4603_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_4604_le__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).

% le_log_of_power
thf(fact_4605_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_4606_log__base__pow,axiom,
    ! [A: real,N: nat,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N ) @ X )
        = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log_base_pow
thf(fact_4607_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_4608_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( semiri5074537144036343181t_real @ N )
        = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% log2_of_power_eq
thf(fact_4609_log__of__power__less,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_less
thf(fact_4610_log__of__power__le,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_le
thf(fact_4611_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_4612_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_4613_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_less
thf(fact_4614_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( N
          = ( suc @ ( suc @ Va2 ) ) )
       => ( ~ ( 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 @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).

% nested_mint
thf(fact_4615_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_4616_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_4617_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_4618_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_4619_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_4620_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_4621_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_4622_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_4623_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 )
       => ( ! [X4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
             => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
          & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_4624_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_4625_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_4626_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_4627_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_4628_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_4629_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_4630_option_Osel,axiom,
    ! [X22: nat] :
      ( ( the_nat @ ( some_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_4631_option_Osel,axiom,
    ! [X22: num] :
      ( ( the_num @ ( some_num @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_4632_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_4633_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_4634_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_4635_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_4636_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_4637_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_4638_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_4639_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_4640_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs2: set_nat,X2: nat,Y5: nat] :
          ( ( member_nat @ Y5 @ Xs2 )
          & ( ord_less_nat @ X2 @ Y5 )
          & ! [Z7: nat] :
              ( ( member_nat @ Z7 @ Xs2 )
             => ( ( ord_less_nat @ X2 @ Z7 )
               => ( ord_less_eq_nat @ Y5 @ Z7 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_4641_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs2: set_nat,X2: nat,Y5: nat] :
          ( ( member_nat @ Y5 @ Xs2 )
          & ( ord_less_nat @ Y5 @ X2 )
          & ! [Z7: nat] :
              ( ( member_nat @ Z7 @ Xs2 )
             => ( ( ord_less_nat @ Z7 @ X2 )
               => ( ord_less_eq_nat @ Z7 @ Y5 ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_4642_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_4643_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_4644_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_4645_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_4646_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N4: nat,TreeList3: list_VEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X2 @ N4 ) ) @ ( vEBT_VEBT_low @ X2 @ N4 ) ) ) ) ).

% in_children_def
thf(fact_4647_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_4648_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_4649_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
        | ( ord_less_nat @ Ma @ X ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_4650_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_4651_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T @ X ) @ N ) ) ).

% delete_pres_valid
thf(fact_4652_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_4653_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( N = one_one_nat )
       => ? [A6: $o,B5: $o] :
            ( T
            = ( vEBT_Leaf @ A6 @ B5 ) ) ) ) ).

% deg_1_Leafy
thf(fact_4654_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A6: $o,B5: $o] :
          ( T
          = ( vEBT_Leaf @ A6 @ B5 ) ) ) ).

% deg_1_Leaf
thf(fact_4655_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A4: $o,B2: $o] :
            ( T
            = ( vEBT_Leaf @ A4 @ B2 ) ) ) ) ).

% deg1Leaf
thf(fact_4656_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).

% maxbmo
thf(fact_4657_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_V8194947554948674370ptions @ T @ Y ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_4658_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_vebt_member @ T @ Y ) ) ) ) ).

% dele_member_cont_corr
thf(fact_4659_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_4660_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_4661_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_4662_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_4663_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_4664_del__single__cont,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_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_4665_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_4666_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_4667_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_4668_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_4669_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B )
          | ( ( ord_less_eq_rat @ C @ A )
            & ( ord_less_eq_rat @ B @ D2 )
            & ( ( ord_less_rat @ C @ A )
              | ( ord_less_rat @ B @ D2 ) ) ) )
        & ( ord_less_eq_rat @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4670_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
      ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_set_nat @ A @ B )
          | ( ( ord_less_eq_set_nat @ C @ A )
            & ( ord_less_eq_set_nat @ B @ D2 )
            & ( ( ord_less_set_nat @ C @ A )
              | ( ord_less_set_nat @ B @ D2 ) ) ) )
        & ( ord_less_eq_set_nat @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4671_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B )
          | ( ( ord_less_eq_num @ C @ A )
            & ( ord_less_eq_num @ B @ D2 )
            & ( ( ord_less_num @ C @ A )
              | ( ord_less_num @ B @ D2 ) ) ) )
        & ( ord_less_eq_num @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4672_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B @ D2 )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B @ D2 ) ) ) )
        & ( ord_less_eq_nat @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4673_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B @ D2 )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B @ D2 ) ) ) )
        & ( ord_less_eq_int @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4674_atLeastatMost__psubset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le1307284697595431911nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or189985376899183464nteger @ C @ D2 ) )
      = ( ( ~ ( ord_le3102999989581377725nteger @ A @ B )
          | ( ( ord_le3102999989581377725nteger @ C @ A )
            & ( ord_le3102999989581377725nteger @ B @ D2 )
            & ( ( ord_le6747313008572928689nteger @ C @ A )
              | ( ord_le6747313008572928689nteger @ B @ D2 ) ) ) )
        & ( ord_le3102999989581377725nteger @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4675_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_real @ A @ B )
          | ( ( ord_less_eq_real @ C @ A )
            & ( ord_less_eq_real @ B @ D2 )
            & ( ( ord_less_real @ C @ A )
              | ( ord_less_real @ B @ D2 ) ) ) )
        & ( ord_less_eq_real @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4676_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N )
           => ( P @ M5 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X2 ) ) ) ) ).

% all_nat_less
thf(fact_4677_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N )
            & ( P @ M5 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X2 ) ) ) ) ).

% ex_nat_less
thf(fact_4678_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_4679_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_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,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
           => ~ ! [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va3 @ Vb @ Vc2 ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
thf(fact_4680_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_4681_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_4682_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A6: $o,B5: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ X3 ) )
     => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S3 ) @ X3 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_4683_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_4684_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A6: $o,B5: $o] :
          ( X
         != ( vEBT_Leaf @ A6 @ B5 ) )
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
thf(fact_4685_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_4686_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N ) ) )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
thf(fact_4687_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N ) ) )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
thf(fact_4688_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_4689_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
thf(fact_4690_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
thf(fact_4691_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_4692_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or4029947393144176647an_rat @ C @ D2 ) )
      = ( ( ord_less_eq_rat @ A @ B )
       => ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_rat @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4693_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or3540276404033026485et_nat @ C @ D2 ) )
      = ( ( ord_less_eq_set_nat @ A @ B )
       => ( ( ord_less_eq_set_nat @ C @ A )
          & ( ord_less_set_nat @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4694_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C @ D2 ) )
      = ( ( ord_less_eq_num @ A @ B )
       => ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_num @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4695_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or66887138388493659n_real @ C @ D2 ) )
      = ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_real @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4696_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
      = ( ( ord_less_eq_nat @ A @ B )
       => ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_nat @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4697_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D2 ) )
      = ( ( ord_less_eq_int @ A @ B )
       => ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_int @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4698_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or8404916559141939852nteger @ C @ D2 ) )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
       => ( ( ord_le3102999989581377725nteger @ C @ A )
          & ( ord_le6747313008572928689nteger @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4699_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D2 ) )
      = ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_4700_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D2 ) )
      = ( ( ord_less_real @ A @ B )
       => ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_4701_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
thf(fact_4702_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
thf(fact_4703_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_space2 @ ( vEBT_Leaf @ A @ B ) )
      = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.space'.simps(1)
thf(fact_4704_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_space @ ( vEBT_Leaf @ A @ B ) )
      = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.space.simps(1)
thf(fact_4705_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o,Uw: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) )
     => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb ) @ X3 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc2 ) @ X3 ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X3 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_4706_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) )
     => ( ! [A6: $o,Uw: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ Uw ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A6: $o,B5: $o,Va: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ Va ) ) ) )
         => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va3 ) @ Vb ) )
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_4707_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,B5: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B5 ) @ zero_zero_nat ) )
     => ( ! [Uv: $o,Uw: $o,N2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) )
       => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va3 ) )
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_4708_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A6: $o,B5: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ X3 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) @ X3 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ X3 ) )
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_4709_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A6: $o,B5: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ X3 ) )
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X3 ) )
       => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) @ X3 ) )
         => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_4710_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A6: $o,B5: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ zero_zero_nat ) )
     => ( ! [A6: $o,B5: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A6: $o,B5: $o,N2: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ N2 ) ) ) )
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Uu ) )
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) @ X3 ) )
             => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) @ X3 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
thf(fact_4711_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] :
      ( ( A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_4712_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_4713_vebt__pred_Osimps_I3_J,axiom,
    ! [B: $o,A: $o,Va2: nat] :
      ( ( B
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_4714_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) @ X )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) ) ).

% vebt_insert.simps(2)
thf(fact_4715_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) ) ).

% vebt_insert.simps(3)
thf(fact_4716_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( ( vEBT_invar_vebt @ A1 @ A22 )
     => ( ( ? [A6: $o,B5: $o] :
              ( A1
              = ( vEBT_Leaf @ A6 @ B5 ) )
         => ( A22
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( ( A22 = Deg2 )
               => ( ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ( vEBT_invar_vebt @ X4 @ N2 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                     => ( ( M3 = N2 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N2 @ M3 ) )
                         => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A1
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( A22 = Deg2 )
                 => ( ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ( vEBT_invar_vebt @ X4 @ N2 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                       => ( ( M3
                            = ( suc @ N2 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N2 @ M3 ) )
                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_invar_vebt @ X4 @ N2 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                         => ( ( M3 = N2 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N2 @ M3 ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X4: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                & ! [X4: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X4 )
                                                      & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X4: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                           => ( vEBT_invar_vebt @ X4 @ N2 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                           => ( ( M3
                                = ( suc @ N2 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N2 @ M3 ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                     => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X4: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                  & ! [X4: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X4 )
                                                        & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_4717_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A12: vEBT_VEBT,A23: nat] :
          ( ( ? [A4: $o,B2: $o] :
                ( A12
                = ( vEBT_Leaf @ A4 @ B2 ) )
            & ( A23
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X2 @ N4 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N4 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
              & ( A23
                = ( plus_plus_nat @ N4 @ N4 ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X2 @ N4 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
              & ( A23
                = ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X2 @ N4 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N4 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
              & ( A23
                = ( plus_plus_nat @ N4 @ N4 ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
                          = I2 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
                      & ! [X2: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X2 @ N4 )
                              = I2 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X2 @ N4 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X2 )
                            & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
              & ! [X2: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X2 @ N4 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N4 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
              & ( A23
                = ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X2: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
                          = I2 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
                      & ! [X2: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X2 @ N4 )
                              = I2 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X2 @ N4 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X2 )
                            & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_4718_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
           => ~ ( ( B5
                 => ( Y
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B5
                 => ( ( A6
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A6
                     => ( 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] :
                ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
               => ( Y
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_4719_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
           => ~ ( ( A6
                 => ( Y
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A6
                 => ( ( B5
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B5
                     => ( 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,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
               => ( Y
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_4720_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X ) )
       => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% minNull_delete_time_bound
thf(fact_4721_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_4722_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_4723_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).

% vebt_delete.simps(6)
thf(fact_4724_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_4725_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).

% not_min_Null_member
thf(fact_4726_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X ) ) ).

% min_Null_member
thf(fact_4727_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_4728_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_4729_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X ) )
       => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) @ one_one_nat ) ) ) ).

% minNull_delete_time_bound'
thf(fact_4730_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_4731_bset_I1_J,axiom,
    ! [D4: int,B3: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ B3 )
                 => ( X3
                   != ( plus_plus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ B3 )
                   => ( X3
                     != ( plus_plus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
             => ( ( P @ ( minus_minus_int @ X4 @ D4 ) )
                & ( Q @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_4732_bset_I2_J,axiom,
    ! [D4: int,B3: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ B3 )
                 => ( X3
                   != ( plus_plus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ B3 )
                   => ( X3
                     != ( plus_plus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
             => ( ( P @ ( minus_minus_int @ X4 @ D4 ) )
                | ( Q @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_4733_aset_I1_J,axiom,
    ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ A2 )
                 => ( X3
                   != ( minus_minus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ A2 )
                   => ( X3
                     != ( minus_minus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
             => ( ( P @ ( plus_plus_int @ X4 @ D4 ) )
                & ( Q @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_4734_aset_I2_J,axiom,
    ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ A2 )
                 => ( X3
                   != ( minus_minus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ A2 )
                   => ( X3
                     != ( minus_minus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
             => ( ( P @ ( plus_plus_int @ X4 @ D4 ) )
                | ( Q @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_4735_aset_I10_J,axiom,
    ! [D2: int,D4: int,A2: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T ) )
           => ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% aset(10)
thf(fact_4736_aset_I9_J,axiom,
    ! [D2: int,D4: int,A2: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T ) )
           => ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% aset(9)
thf(fact_4737_bset_I10_J,axiom,
    ! [D2: int,D4: int,B3: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T ) )
           => ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% bset(10)
thf(fact_4738_bset_I9_J,axiom,
    ! [D2: int,D4: int,B3: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T ) )
           => ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% bset(9)
thf(fact_4739_bset_I3_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X4 = T )
             => ( ( minus_minus_int @ X4 @ D4 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_4740_bset_I4_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B3 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X4 != T )
             => ( ( minus_minus_int @ X4 @ D4 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_4741_bset_I5_J,axiom,
    ! [D4: int,B3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_int @ X4 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_4742_bset_I7_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B3 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_int @ T @ X4 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_4743_aset_I3_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X4 = T )
             => ( ( plus_plus_int @ X4 @ D4 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_4744_aset_I4_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A2 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X4 != T )
             => ( ( plus_plus_int @ X4 @ D4 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_4745_aset_I5_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A2 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_int @ X4 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_4746_aset_I7_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_int @ T @ X4 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_4747_periodic__finite__ex,axiom,
    ! [D2: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K2: int] :
            ( ( P @ X3 )
            = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ( ? [X8: int] : ( P @ X8 ) )
          = ( ? [X2: int] :
                ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
                & ( P @ X2 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_4748_bset_I6_J,axiom,
    ! [D4: int,B3: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_eq_int @ X4 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_4749_bset_I8_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X4 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_4750_aset_I6_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_eq_int @ X4 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_4751_aset_I8_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X4 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_4752_cppi,axiom,
    ! [D4: int,P: int > $o,P2: int > $o,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( P @ X3 )
              = ( P2 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb: int] :
                      ( ( member_int @ Xb @ A2 )
                     => ( X3
                       != ( minus_minus_int @ Xb @ Xa ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P2 @ X3 )
                = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X8: int] : ( P @ X8 ) )
              = ( ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P2 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y5: int] :
                        ( ( member_int @ Y5 @ A2 )
                        & ( P @ ( minus_minus_int @ Y5 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_4753_cpmi,axiom,
    ! [D4: int,P: int > $o,P2: int > $o,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( P @ X3 )
              = ( P2 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb: int] :
                      ( ( member_int @ Xb @ B3 )
                     => ( X3
                       != ( plus_plus_int @ Xb @ Xa ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P2 @ X3 )
                = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X8: int] : ( P @ X8 ) )
              = ( ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P2 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y5: int] :
                        ( ( member_int @ Y5 @ B3 )
                        & ( P @ ( plus_plus_int @ Y5 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_4754_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 ) ) )
       => ~ ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A6: product_prod_nat_nat,B5: product_prod_nat_nat] :
              ( X
             != ( produc2899441246263362727at_nat @ F3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A6 ) @ ( some_P7363390416028606310at_nat @ B5 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_4755_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 ) ) )
       => ~ ! [F3: nat > nat > nat,A6: nat,B5: nat] :
              ( X
             != ( produc8929957630744042906on_nat @ F3 @ ( produc5098337634421038937on_nat @ ( some_nat @ A6 ) @ ( some_nat @ B5 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_4756_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 ) ) )
       => ~ ! [F3: num > num > num,A6: num,B5: num] :
              ( X
             != ( produc5778274026573060048on_num @ F3 @ ( produc8585076106096196333on_num @ ( some_num @ A6 ) @ ( some_num @ B5 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_4757_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 ) ) )
       => ~ ! [F3: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
              ( X
             != ( produc3994169339658061776at_nat @ F3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X3 ) @ ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_4758_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 ) ) )
       => ~ ! [F3: nat > nat > $o,X3: nat,Y3: nat] :
              ( X
             != ( produc4035269172776083154on_nat @ F3 @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_4759_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 ) ) )
       => ~ ! [F3: num > num > $o,X3: num,Y3: num] :
              ( X
             != ( produc3576312749637752826on_num @ F3 @ ( produc8585076106096196333on_num @ ( some_num @ X3 ) @ ( some_num @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_4760_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_4761_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_4762_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_4763_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_4764_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_4765_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_4766_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N ) ) )
      = ( vEBT_Leaf @ A @ B ) ) ).

% vebt_delete.simps(3)
thf(fact_4767_vebt__delete_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
      = ( vEBT_Leaf @ $false @ B ) ) ).

% vebt_delete.simps(1)
thf(fact_4768_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_4769_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_4770_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_4771_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa3: option4927543243414619207at_nat,Xb3: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa3 @ Xb3 )
        = Y )
     => ( ( ( Xa3 = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] :
                ( Xa3
                = ( some_P7363390416028606310at_nat @ V2 ) )
           => ( ( Xb3 = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A6: product_prod_nat_nat] :
                ( ( Xa3
                  = ( some_P7363390416028606310at_nat @ A6 ) )
               => ! [B5: product_prod_nat_nat] :
                    ( ( Xb3
                      = ( some_P7363390416028606310at_nat @ B5 ) )
                   => ( Y
                     != ( some_P7363390416028606310at_nat @ ( X @ A6 @ B5 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_4772_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: num > num > num,Xa3: option_num,Xb3: option_num,Y: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X @ Xa3 @ Xb3 )
        = Y )
     => ( ( ( Xa3 = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] :
                ( Xa3
                = ( some_num @ V2 ) )
           => ( ( Xb3 = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A6: num] :
                ( ( Xa3
                  = ( some_num @ A6 ) )
               => ! [B5: num] :
                    ( ( Xb3
                      = ( some_num @ B5 ) )
                   => ( Y
                     != ( some_num @ ( X @ A6 @ B5 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_4773_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: nat > nat > nat,Xa3: option_nat,Xb3: option_nat,Y: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa3 @ Xb3 )
        = Y )
     => ( ( ( Xa3 = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] :
                ( Xa3
                = ( some_nat @ V2 ) )
           => ( ( Xb3 = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A6: nat] :
                ( ( Xa3
                  = ( some_nat @ A6 ) )
               => ! [B5: nat] :
                    ( ( Xb3
                      = ( some_nat @ B5 ) )
                   => ( Y
                     != ( some_nat @ ( X @ A6 @ B5 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_4774_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
      = ( vEBT_Leaf @ A @ $false ) ) ).

% vebt_delete.simps(2)
thf(fact_4775_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).

% vebt_delete.simps(5)
thf(fact_4776_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_4777_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
thf(fact_4778_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
               => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Newlist
                      = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                   => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
thf(fact_4779_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,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
               => ~ Y )
             => ~ ( ? [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va3 @ Vb @ Vc2 ) )
                 => Y ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_4780_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_m_e_m_b_e_r @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% member_bound_size_univ
thf(fact_4781_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) @ X ) ).

% vebt_member.simps(4)
thf(fact_4782_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ T ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% heigt_uplog_rel
thf(fact_4783_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I @ Y )
      = ( list_u1324408373059187874T_VEBT @ Xs @ I @ Y ) ) ).

% list_update_overwrite
thf(fact_4784_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ I @ Y )
      = ( list_u6098035379799741383_VEBTi @ Xs @ I @ Y ) ) ).

% list_update_overwrite
thf(fact_4785_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) )
      = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% length_list_update
thf(fact_4786_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) )
      = ( size_s7982070591426661849_VEBTi @ Xs ) ) ).

% length_list_update
thf(fact_4787_length__list__update,axiom,
    ! [Xs: list_real,I: nat,X: real] :
      ( ( size_size_list_real @ ( list_update_real @ Xs @ I @ X ) )
      = ( size_size_list_real @ Xs ) ) ).

% length_list_update
thf(fact_4788_length__list__update,axiom,
    ! [Xs: list_o,I: nat,X: $o] :
      ( ( size_size_list_o @ ( list_update_o @ Xs @ I @ X ) )
      = ( size_size_list_o @ Xs ) ) ).

% length_list_update
thf(fact_4789_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_4790_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs @ I @ X ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_list_update
thf(fact_4791_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] :
      ( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_4792_list__update__id,axiom,
    ! [Xs: list_int,I: nat] :
      ( ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_4793_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] :
      ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_4794_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat] :
      ( ( list_u6098035379799741383_VEBTi @ Xs @ I @ ( nth_VEBT_VEBTi @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_4795_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs: list_nat,X: nat] :
      ( ( I != J2 )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J2 )
        = ( nth_nat @ Xs @ J2 ) ) ) ).

% nth_list_update_neq
thf(fact_4796_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs: list_int,X: int] :
      ( ( I != J2 )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J2 )
        = ( nth_int @ Xs @ J2 ) ) ) ).

% nth_list_update_neq
thf(fact_4797_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I != J2 )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J2 )
        = ( nth_VEBT_VEBT @ Xs @ J2 ) ) ) ).

% nth_list_update_neq
thf(fact_4798_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( I != J2 )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ J2 )
        = ( nth_VEBT_VEBTi @ Xs @ J2 ) ) ) ).

% nth_list_update_neq
thf(fact_4799_ceiling__zero,axiom,
    ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_4800_ceiling__zero,axiom,
    ( ( archim7802044766580827645g_real @ zero_zero_real )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_4801_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_4802_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_4803_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_4804_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] :
      ( ( ord_less_eq_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) @ I )
     => ( ( list_u6098035379799741383_VEBTi @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_4805_list__update__beyond,axiom,
    ! [Xs: list_real,I: nat,X: real] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ I )
     => ( ( list_update_real @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_4806_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I )
     => ( ( list_update_o @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_4807_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
     => ( ( list_update_nat @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_4808_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I )
     => ( ( list_update_int @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_4809_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBT,J2: nat,X: vEBT_VEBT] :
      ( ~ ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L @ J2 @ X ) @ I )
        = ( nth_VEBT_VEBT @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_4810_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,J2: nat,X: vEBT_VEBTi] :
      ( ~ ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L @ J2 @ X ) @ I )
        = ( nth_VEBT_VEBTi @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_4811_nth__update__invalid,axiom,
    ! [I: nat,L: list_real,J2: nat,X: real] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( nth_real @ ( list_update_real @ L @ J2 @ X ) @ I )
        = ( nth_real @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_4812_nth__update__invalid,axiom,
    ! [I: nat,L: list_o,J2: nat,X: $o] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( nth_o @ ( list_update_o @ L @ J2 @ X ) @ I )
        = ( nth_o @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_4813_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J2: nat,X: nat] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( nth_nat @ ( list_update_nat @ L @ J2 @ X ) @ I )
        = ( nth_nat @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_4814_nth__update__invalid,axiom,
    ! [I: nat,L: list_int,J2: nat,X: int] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( nth_int @ ( list_update_int @ L @ J2 @ X ) @ I )
        = ( nth_int @ L @ I ) ) ) ).

% nth_update_invalid
thf(fact_4815_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4816_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4817_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_real,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
     => ( ( nth_real @ ( list_update_real @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4818_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( list_update_o @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4819_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4820_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4821_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_4822_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_4823_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_4824_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_4825_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_4826_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_4827_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_4828_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_4829_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_4830_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_4831_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_4832_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_4833_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_4834_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_4835_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_4836_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_4837_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J2: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ J2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J2 ) ) @ J2 @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
          = ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_4838_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J2: nat] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( ord_less_nat @ J2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ ( nth_VEBT_VEBTi @ Xs @ J2 ) ) @ J2 @ ( nth_VEBT_VEBTi @ Xs @ I ) ) )
          = ( set_VEBT_VEBTi2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_4839_set__swap,axiom,
    ! [I: nat,Xs: list_real,J2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_real @ Xs ) )
       => ( ( set_real2 @ ( list_update_real @ ( list_update_real @ Xs @ I @ ( nth_real @ Xs @ J2 ) ) @ J2 @ ( nth_real @ Xs @ I ) ) )
          = ( set_real2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_4840_set__swap,axiom,
    ! [I: nat,Xs: list_o,J2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_o @ Xs ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I @ ( nth_o @ Xs @ J2 ) ) @ J2 @ ( nth_o @ Xs @ I ) ) )
          = ( set_o2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_4841_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I ) ) )
          = ( set_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_4842_set__swap,axiom,
    ! [I: nat,Xs: list_int,J2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J2 ) ) @ J2 @ ( nth_int @ Xs @ I ) ) )
          = ( set_int2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_4843_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_4844_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_4845_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_4846_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_4847_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_4848_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_4849_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_4850_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_4851_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_4852_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_4853_list__update__swap,axiom,
    ! [I: nat,I7: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X6: vEBT_VEBT] :
      ( ( I != I7 )
     => ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I7 @ X6 )
        = ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I7 @ X6 ) @ I @ X ) ) ) ).

% list_update_swap
thf(fact_4854_list__update__swap,axiom,
    ! [I: nat,I7: nat,Xs: list_VEBT_VEBTi,X: vEBT_VEBTi,X6: vEBT_VEBTi] :
      ( ( I != I7 )
     => ( ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ I7 @ X6 )
        = ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I7 @ X6 ) @ I @ X ) ) ) ).

% list_update_swap
thf(fact_4855_set__update__subsetI,axiom,
    ! [Xs: list_complex,A2: set_complex,X: complex,I: nat] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_4856_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,X: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_4857_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,X: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_4858_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_4859_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBTi,A2: set_VEBT_VEBTi,X: vEBT_VEBTi,I: nat] :
      ( ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ Xs ) @ A2 )
     => ( ( member_VEBT_VEBTi @ X @ A2 )
       => ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_4860_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_4861_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_4862_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_4863_in__set__upd__eq,axiom,
    ! [I: nat,L: list_complex,X: complex,Y: complex] :
      ( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ L ) )
     => ( ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_complex @ X @ ( set_complex2 @ L ) )
            & ! [Y5: complex] : ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4864_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L ) )
            & ! [Y5: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4865_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L ) )
            & ! [Y5: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4866_in__set__upd__eq,axiom,
    ! [I: nat,L: list_real,X: real,Y: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( member_real @ X @ ( set_real2 @ ( list_update_real @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_real @ X @ ( set_real2 @ L ) )
            & ! [Y5: real] : ( member_real @ X @ ( set_real2 @ ( list_update_real @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4867_in__set__upd__eq,axiom,
    ! [I: nat,L: list_o,X: $o,Y: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( member_o @ X @ ( set_o2 @ ( list_update_o @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_o @ X @ ( set_o2 @ L ) )
            & ! [Y5: $o] : ( member_o @ X @ ( set_o2 @ ( list_update_o @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4868_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_nat @ X @ ( set_nat2 @ L ) )
            & ! [Y5: nat] : ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4869_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,X: int,Y: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_int @ X @ ( set_int2 @ L ) )
            & ! [Y5: int] : ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y5 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_4870_in__set__upd__cases,axiom,
    ! [X: complex,L: list_complex,I: nat,Y: complex] :
      ( ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ L ) )
         => ( X != Y ) )
       => ( member_complex @ X @ ( set_complex2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4871_in__set__upd__cases,axiom,
    ! [X: vEBT_VEBT,L: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
         => ( X != Y ) )
       => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4872_in__set__upd__cases,axiom,
    ! [X: vEBT_VEBTi,L: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
         => ( X != Y ) )
       => ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4873_in__set__upd__cases,axiom,
    ! [X: real,L: list_real,I: nat,Y: real] :
      ( ( member_real @ X @ ( set_real2 @ ( list_update_real @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
         => ( X != Y ) )
       => ( member_real @ X @ ( set_real2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4874_in__set__upd__cases,axiom,
    ! [X: $o,L: list_o,I: nat,Y: $o] :
      ( ( member_o @ X @ ( set_o2 @ ( list_update_o @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
         => ( X = ~ Y ) )
       => ( member_o @ X @ ( set_o2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4875_in__set__upd__cases,axiom,
    ! [X: nat,L: list_nat,I: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
         => ( X != Y ) )
       => ( member_nat @ X @ ( set_nat2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4876_in__set__upd__cases,axiom,
    ! [X: int,L: list_int,I: nat,Y: int] :
      ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
         => ( X != Y ) )
       => ( member_int @ X @ ( set_int2 @ L ) ) ) ) ).

% in_set_upd_cases
thf(fact_4877_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_complex,X: complex,Y: complex] :
      ( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ L ) )
     => ( ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: complex] : ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4878_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4879_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4880_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_real,X: real,Y: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( member_real @ X @ ( set_real2 @ ( list_update_real @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: real] : ( member_real @ X @ ( set_real2 @ ( list_update_real @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4881_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_o,X: $o,Y: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( member_o @ X @ ( set_o2 @ ( list_update_o @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: $o] : ( member_o @ X @ ( set_o2 @ ( list_update_o @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4882_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: nat] : ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4883_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,X: int,Y: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y5: int] : ( member_int @ X @ ( set_int2 @ ( list_update_int @ L @ I @ Y5 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_4884_set__update__memI,axiom,
    ! [N: nat,Xs: list_complex,X: complex] :
      ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
     => ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4885_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4886_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4887_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,X: real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
     => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4888_set__update__memI,axiom,
    ! [N: nat,Xs: list_o,X: $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4889_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4890_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4891_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ( ( I = J2 )
          & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L @ I @ X ) @ J2 )
          = X ) )
      & ( ~ ( ( I = J2 )
            & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L @ I @ X ) @ J2 )
          = ( nth_VEBT_VEBT @ L @ J2 ) ) ) ) ).

% nth_list_update'
thf(fact_4892_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ( ( I = J2 )
          & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L @ I @ X ) @ J2 )
          = X ) )
      & ( ~ ( ( I = J2 )
            & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L @ I @ X ) @ J2 )
          = ( nth_VEBT_VEBTi @ L @ J2 ) ) ) ) ).

% nth_list_update'
thf(fact_4893_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L: list_real,X: real] :
      ( ( ( ( I = J2 )
          & ( ord_less_nat @ I @ ( size_size_list_real @ L ) ) )
       => ( ( nth_real @ ( list_update_real @ L @ I @ X ) @ J2 )
          = X ) )
      & ( ~ ( ( I = J2 )
            & ( ord_less_nat @ I @ ( size_size_list_real @ L ) ) )
       => ( ( nth_real @ ( list_update_real @ L @ I @ X ) @ J2 )
          = ( nth_real @ L @ J2 ) ) ) ) ).

% nth_list_update'
thf(fact_4894_nth__list__update_H,axiom,
    ! [L: list_o,I: nat,X: $o,J2: nat] :
      ( ( nth_o @ ( list_update_o @ L @ I @ X ) @ J2 )
      = ( ( ( ( I = J2 )
            & ( ord_less_nat @ I @ ( size_size_list_o @ L ) ) )
         => X )
        & ( ~ ( ( I = J2 )
              & ( ord_less_nat @ I @ ( size_size_list_o @ L ) ) )
         => ( nth_o @ L @ J2 ) ) ) ) ).

% nth_list_update'
thf(fact_4895_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L: list_nat,X: nat] :
      ( ( ( ( I = J2 )
          & ( ord_less_nat @ I @ ( size_size_list_nat @ L ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L @ I @ X ) @ J2 )
          = X ) )
      & ( ~ ( ( I = J2 )
            & ( ord_less_nat @ I @ ( size_size_list_nat @ L ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L @ I @ X ) @ J2 )
          = ( nth_nat @ L @ J2 ) ) ) ) ).

% nth_list_update'
thf(fact_4896_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L: list_int,X: int] :
      ( ( ( ( I = J2 )
          & ( ord_less_nat @ I @ ( size_size_list_int @ L ) ) )
       => ( ( nth_int @ ( list_update_int @ L @ I @ X ) @ J2 )
          = X ) )
      & ( ~ ( ( I = J2 )
            & ( ord_less_nat @ I @ ( size_size_list_int @ L ) ) )
       => ( ( nth_int @ ( list_update_int @ L @ I @ X ) @ J2 )
          = ( nth_int @ L @ J2 ) ) ) ) ).

% nth_list_update'
thf(fact_4897_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J2: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( I = J2 )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J2 )
            = X ) )
        & ( ( I != J2 )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J2 )
            = ( nth_VEBT_VEBT @ Xs @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_4898_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J2: nat,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( ( I = J2 )
         => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ J2 )
            = X ) )
        & ( ( I != J2 )
         => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ J2 )
            = ( nth_VEBT_VEBTi @ Xs @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_4899_nth__list__update,axiom,
    ! [I: nat,Xs: list_real,J2: nat,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
     => ( ( ( I = J2 )
         => ( ( nth_real @ ( list_update_real @ Xs @ I @ X ) @ J2 )
            = X ) )
        & ( ( I != J2 )
         => ( ( nth_real @ ( list_update_real @ Xs @ I @ X ) @ J2 )
            = ( nth_real @ Xs @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_4900_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,X: $o,J2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( list_update_o @ Xs @ I @ X ) @ J2 )
        = ( ( ( I = J2 )
           => X )
          & ( ( I != J2 )
           => ( nth_o @ Xs @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_4901_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J2: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I = J2 )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J2 )
            = X ) )
        & ( ( I != J2 )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J2 )
            = ( nth_nat @ Xs @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_4902_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J2: nat,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( I = J2 )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J2 )
            = X ) )
        & ( ( I != J2 )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J2 )
            = ( nth_int @ Xs @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_4903_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
          = Xs )
        = ( ( nth_VEBT_VEBT @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4904_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( ( list_u6098035379799741383_VEBTi @ Xs @ I @ X )
          = Xs )
        = ( ( nth_VEBT_VEBTi @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4905_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_real,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
     => ( ( ( list_update_real @ Xs @ I @ X )
          = Xs )
        = ( ( nth_real @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4906_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ( list_update_o @ Xs @ I @ X )
          = Xs )
        = ( ( nth_o @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4907_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I @ X )
          = Xs )
        = ( ( nth_nat @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4908_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( list_update_int @ Xs @ I @ X )
          = Xs )
        = ( ( nth_int @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4909_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_4910_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_4911_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: vEBT_VEBT > vEBT_VEBT > assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ I5 @ A2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( vEBT_L3204528365124325536T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4912_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: vEBT_VEBTi > vEBT_VEBT > assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ I5 @ A2 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( vEBT_L2497118539674116125T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4913_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: vEBT_VEBTi > vEBT_VEBTi > assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ I5 @ A2 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( vEBT_L886525131989349516_VEBTi @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4914_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: real > vEBT_VEBT > assn,X1: real,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ I5 @ A2 @ ( list_update_real @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( vEBT_L3095048238742455910T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4915_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: real > vEBT_VEBTi > assn,X1: real,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ I5 @ A2 @ ( list_update_real @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( vEBT_L7851252805511451907_VEBTi @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4916_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_o,A2: $o > vEBT_VEBT > assn,X1: $o,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( vEBT_L1319876754960170684T_VEBT @ I5 @ A2 @ ( list_update_o @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( vEBT_L1319876754960170684T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4917_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_o,A2: $o > vEBT_VEBTi > assn,X1: $o,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( vEBT_L6286945158656146733_VEBTi @ I5 @ A2 @ ( list_update_o @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( vEBT_L6286945158656146733_VEBTi @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4918_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A2: nat > vEBT_VEBT > assn,X1: nat,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( vEBT_L8511957252848910786T_VEBT @ I5 @ A2 @ ( list_update_nat @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( vEBT_L8511957252848910786T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4919_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A2: nat > vEBT_VEBTi > assn,X1: nat,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( vEBT_L7489483478785760935_VEBTi @ I5 @ A2 @ ( list_update_nat @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( vEBT_L7489483478785760935_VEBTi @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4920_subst__not__in,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A2: int > vEBT_VEBT > assn,X1: int,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L2018189785592951398T_VEBT @ I5 @ A2 @ ( list_update_int @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( vEBT_L2018189785592951398T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_4921_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_4922_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_4923_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_4924_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_4925_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_4926_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_4927_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
thf(fact_4928_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( 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 @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_4929_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_4930_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_4931_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_4932_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X ) ).

% vebt_member.simps(3)
thf(fact_4933_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 ) )
         => ~ ! [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va3 @ Vb @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_4934_member__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ).

% member_bound_height
thf(fact_4935_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_4936_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_4937_log__ceil__idem,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ) ) ).

% log_ceil_idem
thf(fact_4938_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_s_u_c_c @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% succ_bound_size_univ
thf(fact_4939_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_p_r_e_d @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% pred_bound_size_univ
thf(fact_4940_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_i_n_s_e_r_t @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% insert_bound_size_univ
thf(fact_4941_lemma__termdiff3,axiom,
    ! [H2: real,Z: real,K6: real,N: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K6 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K6 )
         => ( 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 @ K6 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_4942_lemma__termdiff3,axiom,
    ! [H2: complex,Z: complex,K6: real,N: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K6 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K6 )
         => ( 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 @ K6 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_4943_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ( ring_1_of_int_real @ W )
        = ( ring_1_of_int_real @ Z ) )
      = ( W = Z ) ) ).

% of_int_eq_iff
thf(fact_4944_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ( ring_1_of_int_rat @ W )
        = ( ring_1_of_int_rat @ Z ) )
      = ( W = Z ) ) ).

% of_int_eq_iff
thf(fact_4945_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ( ring_17405671764205052669omplex @ W )
        = ( ring_17405671764205052669omplex @ Z ) )
      = ( W = Z ) ) ).

% of_int_eq_iff
thf(fact_4946_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_int_of_bool
thf(fact_4947_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% of_int_of_bool
thf(fact_4948_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_17405671764205052669omplex @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n1201886186963655149omplex @ P ) ) ).

% of_int_of_bool
thf(fact_4949_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_int_of_bool
thf(fact_4950_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_18347121197199848620nteger @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_int_of_bool
thf(fact_4951_of__int__ceiling__cancel,axiom,
    ! [X: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
        = X )
      = ( ? [N4: int] :
            ( X
            = ( ring_1_of_int_rat @ N4 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_4952_of__int__ceiling__cancel,axiom,
    ! [X: real] :
      ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
        = X )
      = ( ? [N4: int] :
            ( X
            = ( ring_1_of_int_real @ N4 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_4953_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).

% max_Suc_Suc
thf(fact_4954_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_4955_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_4956_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_4957_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_4958_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_4959_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_4960_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_4961_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_4962_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_4963_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X ) )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(3)
thf(fact_4964_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_4965_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_4966_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_4967_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_4968_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_4969_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ zero_z3403309356797280102nteger )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(4)
thf(fact_4970_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_4971_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_4972_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_4973_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_4974_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_4975_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_4976_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_4977_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_4978_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_4979_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_4980_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = zero_zero_int )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4981_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_18347121197199848620nteger @ Z )
        = zero_z3403309356797280102nteger )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4982_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_real @ Z )
        = zero_zero_real )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4983_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = zero_zero_rat )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4984_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = zero_zero_complex )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_4985_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int
        = ( ring_1_of_int_int @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4986_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_z3403309356797280102nteger
        = ( ring_18347121197199848620nteger @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4987_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real
        = ( ring_1_of_int_real @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4988_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat
        = ( ring_1_of_int_rat @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4989_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_complex
        = ( ring_17405671764205052669omplex @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_4990_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_4991_of__int__0,axiom,
    ( ( ring_18347121197199848620nteger @ zero_zero_int )
    = zero_z3403309356797280102nteger ) ).

% of_int_0
thf(fact_4992_of__int__0,axiom,
    ( ( ring_1_of_int_real @ zero_zero_int )
    = zero_zero_real ) ).

% of_int_0
thf(fact_4993_of__int__0,axiom,
    ( ( ring_1_of_int_rat @ zero_zero_int )
    = zero_zero_rat ) ).

% of_int_0
thf(fact_4994_of__int__0,axiom,
    ( ( ring_17405671764205052669omplex @ zero_zero_int )
    = zero_zero_complex ) ).

% of_int_0
thf(fact_4995_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_4996_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_4997_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_4998_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_4999_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
      = ( numera6690914467698888265omplex @ K ) ) ).

% of_int_numeral
thf(fact_5000_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_5001_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_5002_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_5003_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_5004_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_5005_max__0__1_I2_J,axiom,
    ( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
    = one_one_Code_integer ) ).

% max_0_1(2)
thf(fact_5006_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_5007_max__0__1_I2_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(2)
thf(fact_5008_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_5009_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_5010_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_5011_max__0__1_I1_J,axiom,
    ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% max_0_1(1)
thf(fact_5012_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_5013_max__0__1_I1_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(1)
thf(fact_5014_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_5015_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_5016_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_5017_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_5018_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_5019_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_5020_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_5021_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_5022_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_5023_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_5024_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_5025_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_5026_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_5027_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_5028_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_5029_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_5030_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_5031_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_5032_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_5033_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_5034_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_5035_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_5036_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_5037_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_5038_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_5039_of__int__1,axiom,
    ( ( ring_17405671764205052669omplex @ one_one_int )
    = one_one_complex ) ).

% of_int_1
thf(fact_5040_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_5041_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_5042_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_5043_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_17405671764205052669omplex @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_int_add
thf(fact_5044_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_17405671764205052669omplex @ ( times_times_int @ W @ Z ) )
      = ( times_times_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_int_mult
thf(fact_5045_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_5046_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_5047_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_5048_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_17405671764205052669omplex @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_int_diff
thf(fact_5049_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_5050_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_5051_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_5052_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% of_int_of_nat_eq
thf(fact_5053_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% of_int_of_nat_eq
thf(fact_5054_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% of_int_of_nat_eq
thf(fact_5055_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_17405671764205052669omplex @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri8010041392384452111omplex @ N ) ) ).

% of_int_of_nat_eq
thf(fact_5056_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_5057_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_5058_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_5059_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_18347121197199848620nteger @ X )
        = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_5060_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_5061_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_5062_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_5063_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_5064_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W )
        = ( ring_18347121197199848620nteger @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_5065_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_5066_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_5067_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_5068_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_5069_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_18347121197199848620nteger @ ( power_power_int @ Z @ N ) )
      = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ Z ) @ N ) ) ).

% of_int_power
thf(fact_5070_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_5071_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_5072_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_5073_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_5074_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_5075_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_5076_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_5077_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ zero_z3403309356797280102nteger )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5078_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_5079_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_5080_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_5081_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_5082_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_5083_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_5084_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_5085_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_5086_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_5087_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_5088_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_5089_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_5090_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ zero_z3403309356797280102nteger )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5091_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_5092_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_5093_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_5094_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_5095_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5096_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_5097_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_5098_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_5099_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5100_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_5101_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_5102_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_5103_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_5104_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_5105_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_5106_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_5107_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_5108_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_5109_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_5110_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_5111_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_5112_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5113_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_5114_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_5115_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_5116_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5117_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_5118_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_5119_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_5120_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_5121_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_5122_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_5123_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_5124_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_5125_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_5126_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_5127_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5128_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_5129_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_5130_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_5131_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) @ ( ring_18347121197199848620nteger @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5132_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_5133_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_5134_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_5135_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_5136_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_5137_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_5138_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5139_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_5140_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_5141_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_5142_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B ) @ W ) @ ( ring_18347121197199848620nteger @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5143_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5144_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_5145_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_5146_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_5147_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5148_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_5149_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_5150_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_5151_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5152_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_5153_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_5154_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_5155_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5156_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_5157_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_5158_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_5159_of__int__max,axiom,
    ! [X: int,Y: int] :
      ( ( ring_1_of_int_real @ ( ord_max_int @ X @ Y ) )
      = ( ord_max_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ Y ) ) ) ).

% of_int_max
thf(fact_5160_of__int__max,axiom,
    ! [X: int,Y: int] :
      ( ( ring_1_of_int_rat @ ( ord_max_int @ X @ Y ) )
      = ( ord_max_rat @ ( ring_1_of_int_rat @ X ) @ ( ring_1_of_int_rat @ Y ) ) ) ).

% of_int_max
thf(fact_5161_of__int__max,axiom,
    ! [X: int,Y: int] :
      ( ( ring_1_of_int_int @ ( ord_max_int @ X @ Y ) )
      = ( ord_max_int @ ( ring_1_of_int_int @ X ) @ ( ring_1_of_int_int @ Y ) ) ) ).

% of_int_max
thf(fact_5162_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_5163_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_5164_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_5165_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_5166_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_5167_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_5168_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_5169_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_5170_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_5171_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_5172_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_5173_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_5174_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_5175_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_5176_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_5177_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z6: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z6 ) @ X ) ).

% ex_of_int_less
thf(fact_5178_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z6: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z6 ) @ X ) ).

% ex_of_int_less
thf(fact_5179_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z6: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z6 ) ) ).

% ex_less_of_int
thf(fact_5180_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z6: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z6 ) ) ).

% ex_less_of_int
thf(fact_5181_mult__of__int__commute,axiom,
    ! [X: int,Y: complex] :
      ( ( times_times_complex @ ( ring_17405671764205052669omplex @ X ) @ Y )
      = ( times_times_complex @ Y @ ( ring_17405671764205052669omplex @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5182_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_5183_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_5184_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_5185_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).

% nat_add_max_right
thf(fact_5186_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).

% nat_add_max_left
thf(fact_5187_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_max_right
thf(fact_5188_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).

% nat_mult_max_left
thf(fact_5189_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
      = ( ord_max_nat @ N @ M ) ) ).

% nat_minus_add_max
thf(fact_5190_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_5191_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_5192_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_5193_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_5194_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_5195_real__of__int__div,axiom,
    ! [D2: int,N: int] :
      ( ( dvd_dvd_int @ D2 @ N )
     => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D2 ) )
        = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D2 ) ) ) ) ).

% real_of_int_div
thf(fact_5196_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).

% of_int_nonneg
thf(fact_5197_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_5198_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_5199_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_5200_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_5201_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_5202_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_5203_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).

% of_int_pos
thf(fact_5204_floor__exists,axiom,
    ! [X: rat] :
    ? [Z6: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z6 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z6 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_5205_floor__exists,axiom,
    ! [X: real] :
    ? [Z6: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z6 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z6 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_5206_floor__exists1,axiom,
    ! [X: rat] :
    ? [X3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
thf(fact_5207_floor__exists1,axiom,
    ! [X: real] :
    ? [X3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X )
            & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
thf(fact_5208_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_5209_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_5210_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_5211_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_5212_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_5213_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( ring_18347121197199848620nteger @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5214_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_5215_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_5216_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N4: int,M5: int] : ( ord_less_real @ ( ring_1_of_int_real @ N4 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M5 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_5217_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N4: int,M5: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N4 ) @ one_one_real ) @ ( ring_1_of_int_real @ M5 ) ) ) ) ).

% int_less_real_le
thf(fact_5218_real__of__int__div__aux,axiom,
    ! [X: int,D2: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D2 ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D2 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D2 ) ) @ ( ring_1_of_int_real @ D2 ) ) ) ) ).

% real_of_int_div_aux
thf(fact_5219_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_5220_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
thf(fact_5221_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
thf(fact_5222_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I2 ) ) )
           => ( P @ I2 ) ) ) ) ).

% ceiling_split
thf(fact_5223_ceiling__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
           => ( P @ I2 ) ) ) ) ).

% ceiling_split
thf(fact_5224_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_5225_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_5226_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_5227_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_5228_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_5229_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_5230_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_5231_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_5232_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_5233_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_5234_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_5235_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_5236_ceiling__divide__upper,axiom,
    ! [Q2: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_eq_rat @ P4 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q2 ) ) ) @ Q2 ) ) ) ).

% ceiling_divide_upper
thf(fact_5237_ceiling__divide__upper,axiom,
    ! [Q2: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_eq_real @ P4 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q2 ) ) ) @ Q2 ) ) ) ).

% ceiling_divide_upper
thf(fact_5238_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_5239_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_5240_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va2: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
thf(fact_5241_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
thf(fact_5242_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
thf(fact_5243_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu2: $o,B: $o] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat )
      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
thf(fact_5244_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
thf(fact_5245_ceiling__divide__lower,axiom,
    ! [Q2: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) @ P4 ) ) ).

% ceiling_divide_lower
thf(fact_5246_ceiling__divide__lower,axiom,
    ! [Q2: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) @ P4 ) ) ).

% ceiling_divide_lower
thf(fact_5247_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_5248_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_5249_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
thf(fact_5250_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
thf(fact_5251_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
thf(fact_5252_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_5253_insersimp,axiom,
    ! [T: vEBT_VEBT,N: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ Y ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insersimp
thf(fact_5254_insertsimp,axiom,
    ! [T: vEBT_VEBT,N: nat,L: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_minNull @ T )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ L ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insertsimp
thf(fact_5255_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% insert_bound_height
thf(fact_5256_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% pred_bound_height
thf(fact_5257_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% succ_bound_height
thf(fact_5258_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_5259_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_5260_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_5261_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_5262_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_5263_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_5264_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_5265_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_5266_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
      = ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_5267_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
      = ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_5268_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_5269_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_5270_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
      = Q2 ) ).

% max_enat_simps(2)
thf(fact_5271_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
      = Q2 ) ).

% max_enat_simps(3)
thf(fact_5272_norm__zero,axiom,
    ( ( real_V7735802525324610683m_real @ zero_zero_real )
    = zero_zero_real ) ).

% norm_zero
thf(fact_5273_norm__zero,axiom,
    ( ( real_V1022390504157884413omplex @ zero_zero_complex )
    = zero_zero_real ) ).

% norm_zero
thf(fact_5274_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( ( real_V7735802525324610683m_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_eq_zero
thf(fact_5275_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( ( real_V1022390504157884413omplex @ X )
        = zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_eq_zero
thf(fact_5276_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_5277_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_5278_norm__not__less__zero,axiom,
    ! [X: complex] :
      ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).

% norm_not_less_zero
thf(fact_5279_norm__ge__zero,axiom,
    ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).

% norm_ge_zero
thf(fact_5280_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_5281_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_5282_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_5283_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_5284_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_5285_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_5286_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_5287_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_5288_norm__mult__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).

% norm_mult_less
thf(fact_5289_norm__mult__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).

% norm_mult_less
thf(fact_5290_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_5291_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_5292_norm__add__less,axiom,
    ! [X: real,R2: real,Y: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_5293_norm__add__less,axiom,
    ! [X: complex,R2: real,Y: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_5294_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_5295_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_5296_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_5297_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_5298_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_5299_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_5300_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_5301_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_5302_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_5303_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_5304_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_5305_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_5306_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_5307_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_5308_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_5309_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_5310_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_5311_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_5312_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_5313_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_5314_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_5315_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_5316_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_5317_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_5318_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_5319_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_5320_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D2 ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D2 ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_5321_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D2 ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D2 ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_5322_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_5323_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_5324_norm__power__diff,axiom,
    ! [Z: real,W: real,M: 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 @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_5325_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: 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 @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_5326_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_5327_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_5328_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_5329_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_5330_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_5331_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_5332_max__less__iff__conj,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
      = ( ( ord_le6747313008572928689nteger @ X @ Z )
        & ( ord_le6747313008572928689nteger @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_5333_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_5334_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_5335_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_5336_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_5337_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_5338_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_5339_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_max_Code_integer @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_5340_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_5341_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_5342_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_5343_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_5344_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_5345_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_5346_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_max_Code_integer @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_5347_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ 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_pred @ ( 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_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_5348_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( vEBT_vebt_pred @ ( 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_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ 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_pred @ ( 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_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_5349_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_5350_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_5351_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
               => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                  = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                    @ ( vEBT_Node
                      @ ( some_P7363390416028606310at_nat
                        @ ( product_Pair_nat_nat @ Mi
                          @ ( if_nat @ ( X = Ma )
                            @ ( if_nat
                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                = none_nat )
                              @ Mi
                              @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                            @ Ma ) ) )
                      @ Deg
                      @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                      @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                    @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
thf(fact_5352_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
               => ( ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Sn
                      = ( vEBT_vebt_delete @ Summary @ H2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ Sn )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
thf(fact_5353_del__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
               => ( ( Newlist
                    = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                   => ( ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                      & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
thf(fact_5354_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_x_mia
thf(fact_5355_del__x__mi__lets__in__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( Sn
                            = ( vEBT_vebt_delete @ Summary @ H2 ) )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ Sn )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
thf(fact_5356_del__x__mi__lets__in,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                        & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
thf(fact_5357_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                    = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                      @ ( vEBT_Node
                        @ ( some_P7363390416028606310at_nat
                          @ ( product_Pair_nat_nat @ Xn
                            @ ( if_nat @ ( Xn = Ma )
                              @ ( if_nat
                                @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                  = none_nat )
                                @ Xn
                                @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                              @ Ma ) ) )
                        @ Deg
                        @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                        @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
thf(fact_5358_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_in_range
thf(fact_5359_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_5360_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ A ) )
        @ ( collect_complex
          @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ B ) ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5361_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
        @ ( collect_int
          @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5362_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
        @ ( collect_nat
          @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5363_lambda__zero,axiom,
    ( ( ^ [H: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_5364_lambda__zero,axiom,
    ( ( ^ [H: code_integer] : zero_z3403309356797280102nteger )
    = ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger ) ) ).

% lambda_zero
thf(fact_5365_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_5366_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_5367_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_5368_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_5369_lambda__one,axiom,
    ( ( ^ [X2: assn] : X2 )
    = ( times_times_assn @ one_one_assn ) ) ).

% lambda_one
thf(fact_5370_lambda__one,axiom,
    ( ( ^ [X2: real] : X2 )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_5371_lambda__one,axiom,
    ( ( ^ [X2: rat] : X2 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_5372_lambda__one,axiom,
    ( ( ^ [X2: nat] : X2 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_5373_lambda__one,axiom,
    ( ( ^ [X2: int] : X2 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_5374_mult__commute__abs,axiom,
    ! [C: real] :
      ( ( ^ [X2: real] : ( times_times_real @ X2 @ C ) )
      = ( times_times_real @ C ) ) ).

% mult_commute_abs
thf(fact_5375_mult__commute__abs,axiom,
    ! [C: rat] :
      ( ( ^ [X2: rat] : ( times_times_rat @ X2 @ C ) )
      = ( times_times_rat @ C ) ) ).

% mult_commute_abs
thf(fact_5376_mult__commute__abs,axiom,
    ! [C: nat] :
      ( ( ^ [X2: nat] : ( times_times_nat @ X2 @ C ) )
      = ( times_times_nat @ C ) ) ).

% mult_commute_abs
thf(fact_5377_mult__commute__abs,axiom,
    ! [C: int] :
      ( ( ^ [X2: int] : ( times_times_int @ X2 @ C ) )
      = ( times_times_int @ C ) ) ).

% mult_commute_abs
thf(fact_5378_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_set_complex
        @ ( collect_complex
          @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ A ) )
        @ ( collect_complex
          @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ B ) ) )
      = ( ( dvd_dvd_complex @ A @ B )
        & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5379_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
        @ ( collect_nat
          @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5380_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
        @ ( collect_int
          @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
      = ( ( dvd_dvd_int @ A @ B )
        & ~ ( dvd_dvd_int @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5381_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_5382_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_code(2)
thf(fact_5383_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_5384_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_5385_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_5386_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_5387_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_less_as_int
thf(fact_5388_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_5389_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_5390_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_5391_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_5392_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_5393_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_5394_power__numeral__even,axiom,
    ! [Z: code_integer,W: num] :
      ( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_5395_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_5396_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_5397_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_5398_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_5399_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_5400_power__numeral__odd,axiom,
    ! [Z: code_integer,W: num] :
      ( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ Z @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_5401_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_5402_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_5403_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_5404_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_5405_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X2: nat] :
          ( collect_nat
          @ ^ [N4: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_5406_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
thf(fact_5407_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
thf(fact_5408_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
         => ~ ! [Va: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
thf(fact_5409_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
         => ~ ! [Va: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
thf(fact_5410_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: 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 @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_5411_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_5412_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_5413_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_5414_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_5415_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_5416_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_5417_less__max__iff__disj,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z @ ( ord_max_Code_integer @ X @ Y ) )
      = ( ( ord_le6747313008572928689nteger @ Z @ X )
        | ( ord_le6747313008572928689nteger @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5418_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_5419_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_5420_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_5421_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_5422_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_5423_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_5424_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
     => ~ ( ( ord_le6747313008572928689nteger @ B @ A )
         => ~ ( ord_le6747313008572928689nteger @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5425_max_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B2: extended_enat,A4: extended_enat] :
          ( ( A4
            = ( ord_ma741700101516333627d_enat @ A4 @ B2 ) )
          & ( A4 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5426_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A4: real] :
          ( ( A4
            = ( ord_max_real @ A4 @ B2 ) )
          & ( A4 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5427_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A4: rat] :
          ( ( A4
            = ( ord_max_rat @ A4 @ B2 ) )
          & ( A4 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5428_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A4: num] :
          ( ( A4
            = ( ord_max_num @ A4 @ B2 ) )
          & ( A4 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5429_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A4: nat] :
          ( ( A4
            = ( ord_max_nat @ A4 @ B2 ) )
          & ( A4 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5430_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A4: int] :
          ( ( A4
            = ( ord_max_int @ A4 @ B2 ) )
          & ( A4 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5431_max_Ostrict__order__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B2: code_integer,A4: code_integer] :
          ( ( A4
            = ( ord_max_Code_integer @ A4 @ B2 ) )
          & ( A4 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5432_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_5433_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_5434_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_5435_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_5436_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_5437_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_5438_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ A )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5439_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_5440_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_5441_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_5442_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_5443_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_5444_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_5445_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ B )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5446_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ( vEBT_vebt_member @ X @ Xa3 )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
           => ~ ( ( ( Xa3 = zero_zero_nat )
                 => A6 )
                & ( ( Xa3 != zero_zero_nat )
                 => ( ( ( Xa3 = one_one_nat )
                     => B5 )
                    & ( Xa3 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
             => ~ ( ( Xa3 != Mi2 )
                 => ( ( Xa3 != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_5447_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X = Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X = Ma ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
thf(fact_5448_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: 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 @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_5449_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa3 )
        = Y )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
           => ( Y
              = ( ~ ( ( ( Xa3 = zero_zero_nat )
                     => A6 )
                    & ( ( Xa3 != zero_zero_nat )
                     => ( ( ( Xa3 = one_one_nat )
                         => B5 )
                        & ( Xa3 = 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,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
             => Y )
           => ( ( ? [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) )
               => Y )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                      = ( ~ ( ( Xa3 != Mi2 )
                           => ( ( Xa3 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_5450_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa3 )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
           => ( ( ( Xa3 = zero_zero_nat )
               => A6 )
              & ( ( Xa3 != zero_zero_nat )
               => ( ( ( Xa3 = one_one_nat )
                   => B5 )
                  & ( Xa3 = 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,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
           => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Xa3 != Mi2 )
                     => ( ( Xa3 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_5451_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa3 )
        = Y )
     => ( ( ? [A6: $o,B5: $o] :
              ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
         => ( Y
           != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa3 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
             => ( Y
               != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) )
               => ( Y
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa3 = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa3 = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa3 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
thf(fact_5452_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa3 )
        = Y )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
           => ~ ( ( ( Xa3 = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B5 ) ) )
                & ( ( Xa3 != zero_zero_nat )
                 => ( ( ( Xa3 = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A6 @ $true ) ) )
                    & ( ( Xa3 != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A6 @ B5 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
             => ( Y
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
               => ( Y
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) ) )
           => ( ! [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 @ Xa3 @ Xa3 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                          & ~ ( ( Xa3 = Mi2 )
                              | ( Xa3 = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Xa3 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_5453_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X @ Mi )
          | ( ord_less_nat @ Ma @ X ) )
       => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = one_one_nat ) )
      & ( ~ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
              = one_one_nat ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
              = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( plus_plus_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) @ one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
thf(fact_5454_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: 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 @ Va2 ) ) @ 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 @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_5455_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_5456_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X @ Mi )
          | ( ord_less_nat @ Ma @ X ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
      & ( ~ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ Summary ) )
                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_5457_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa3 )
        = Y )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
           => ( ( Xa3 = zero_zero_nat )
             => ( Y
               != ( vEBT_Leaf @ $false @ B5 ) ) ) )
       => ( ! [A6: $o] :
              ( ? [B5: $o] :
                  ( X
                  = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Xa3
                  = ( suc @ zero_zero_nat ) )
               => ( Y
                 != ( vEBT_Leaf @ A6 @ $false ) ) ) )
         => ( ! [A6: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A6 @ B5 ) )
               => ( ? [N2: nat] :
                      ( Xa3
                      = ( suc @ ( suc @ N2 ) ) )
                 => ( Y
                   != ( vEBT_Leaf @ A6 @ B5 ) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                   => ( Y
                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                     => ( Y
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa3 @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa3 ) )
                             => ( Y
                                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa3 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa3 ) )
                             => ( ( ( ( Xa3 = Mi2 )
                                    & ( Xa3 = Ma2 ) )
                                 => ( Y
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                                & ( ~ ( ( Xa3 = Mi2 )
                                      & ( Xa3 = Ma2 ) )
                                 => ( Y
                                    = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                      @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa3 = Mi2 ) @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa3 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa3 != Mi2 )
                                                   => ( Xa3 = Ma2 ) ) )
                                                @ ( if_nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    = none_nat )
                                                  @ ( if_nat @ ( Xa3 = Mi2 ) @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ Mi2 )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa3 = Mi2 ) @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa3 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa3 != Mi2 )
                                                   => ( Xa3 = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ Summary2 ) )
                                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_5458_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa3 )
        = Y )
     => ( ( ? [A6: $o,B5: $o] :
              ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
         => ( ( Xa3 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A6: $o,B5: $o] :
                ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
           => ( ( Xa3
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A6: $o,B5: $o] :
                  ( X
                  = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ? [N2: nat] :
                    ( Xa3
                    = ( suc @ ( suc @ N2 ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa3 @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa3 ) )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ( ord_less_nat @ Xa3 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa3 ) )
                             => ( ( ( ( Xa3 = Mi2 )
                                    & ( Xa3 = Ma2 ) )
                                 => ( Y = one_one_nat ) )
                                & ( ~ ( ( Xa3 = Mi2 )
                                      & ( Xa3 = Ma2 ) )
                                 => ( Y
                                    = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
thf(fact_5459_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa3 )
        = Y )
     => ( ! [Uu: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ Uu @ B5 ) )
           => ( ( Xa3 = zero_zero_nat )
             => ~ ( ( B5
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B5
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ Uv @ Uw ) )
           => ( ? [N2: nat] :
                  ( Xa3
                  = ( suc @ N2 ) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) )
             => ( 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,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa3 @ Mi2 )
                           => ( Y
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                           => ( Y
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( 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 @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                      = none_nat )
                                    @ none_nat
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_5460_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa3 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( ( Xa3 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A6: $o] :
              ( ? [Uw: $o] :
                  ( X
                  = ( vEBT_Leaf @ A6 @ Uw ) )
             => ( ( Xa3
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A6
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A6
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A6: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A6 @ B5 ) )
               => ( ? [Va: nat] :
                      ( Xa3
                      = ( suc @ ( suc @ Va ) ) )
                 => ~ ( ( B5
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B5
                       => ( ( A6
                           => ( Y
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A6
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va3 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa3 )
                             => ( Y
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( 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 @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa3 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_5461_div__half__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( ( product_Pair_nat_nat @ ( divide_divide_nat @ X @ Y ) @ ( modulo_modulo_nat @ X @ Y ) )
        = ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ Y @ ( minus_minus_nat @ X @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ Y ) ) ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( minus_minus_nat @ X @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ ( minus_minus_nat @ X @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ Y ) ) ) ) ) ) ).

% div_half_nat
thf(fact_5462_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X @ Xa3 )
        = Y )
     => ( ( ? [Uu: $o,B5: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ B5 ) )
         => ( ( Xa3 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [Uv: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ Uv @ Uw ) )
           => ( ? [N2: nat] :
                  ( Xa3
                  = ( suc @ N2 ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) )
             => ( Y != one_one_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 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa3 @ Mi2 )
                           => ( Y = one_one_nat ) )
                          & ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                           => ( Y
                              = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                @ ( if_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
thf(fact_5463_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X @ Xa3 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( ( Xa3 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A6: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ A6 @ Uw ) )
           => ( ( Xa3
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A6: $o,B5: $o] :
                  ( X
                  = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ? [Va: nat] :
                    ( Xa3
                    = ( suc @ ( suc @ Va ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va3 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa3 )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                             => ( Y
                                = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                  @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
thf(fact_5464_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa3 )
        = Y )
     => ( ( ? [A6: $o,B5: $o] :
              ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
         => ( Y
           != ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa3 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
               => ( Y
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
                        @ ( if_nat
                          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                            & ~ ( ( Xa3 = Mi2 )
                                | ( Xa3 = Ma2 ) ) )
                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                          @ one_one_nat ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
thf(fact_5465_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
            @ one_one_nat ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
thf(fact_5466_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa3 )
        = Y )
     => ( ( ? [A6: $o,B5: $o] :
              ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ one_one_nat
                        @ ( if_nat @ ( Xa3 = Mi2 ) @ zero_zero_nat
                          @ ( if_nat @ ( Xa3 = Ma2 ) @ zero_zero_nat
                            @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa3 ) @ zero_zero_nat
                                @ ( if_nat
                                  @ ( ( ord_less_nat @ Mi2 @ Xa3 )
                                    & ( ord_less_nat @ Xa3 @ Ma2 ) )
                                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                                  @ zero_zero_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
thf(fact_5467_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,Va2: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
thf(fact_5468_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
thf(fact_5469_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
thf(fact_5470_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,B: $o] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
thf(fact_5471_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
thf(fact_5472_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
thf(fact_5473_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
thf(fact_5474_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
thf(fact_5475_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_5476_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
thf(fact_5477_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
thf(fact_5478_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% pred_bound_height'
thf(fact_5479_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% succ'_bound_height
thf(fact_5480_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uv: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Uu: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu @ $true ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
               => ( Y != one_one_nat ) )
             => ~ ( ? [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va3 @ Vb @ Vc2 ) )
                 => ( Y != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
thf(fact_5481_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_5482_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% member_bound_height'
thf(fact_5483_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_p_r_e_d2 @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% pred_bound_size_univ'
thf(fact_5484_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_s_u_c_c2 @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% succ_bound_size_univ'
thf(fact_5485_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( X = Mi ) @ zero_zero_nat
          @ ( if_nat @ ( X = Ma ) @ zero_zero_nat
            @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ zero_zero_nat
              @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ zero_zero_nat
                @ ( if_nat
                  @ ( ( ord_less_nat @ Mi @ X )
                    & ( ord_less_nat @ X @ Ma ) )
                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                  @ zero_zero_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
thf(fact_5486_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
        @ ( if_nat
          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ~ ( ( X = Mi )
                | ( X = Ma ) ) )
          @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
          @ one_one_nat ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
thf(fact_5487_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
            @ one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
thf(fact_5488_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa3 )
        = Y )
     => ( ( ? [A6: $o,B5: $o] :
              ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                   => ( Y
                     != ( if_nat
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                          & ~ ( ( Xa3 = Mi2 )
                              | ( Xa3 = Ma2 ) ) )
                        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                        @ one_one_nat ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
thf(fact_5489_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( if_nat
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X = Mi )
              | ( X = Ma ) ) )
        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
        @ one_one_nat ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
thf(fact_5490_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa3 )
        = Y )
     => ( ( ? [A6: $o,B5: $o] :
              ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
         => ( ( Xa3 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A6: $o,B5: $o] :
                ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
           => ( ( Xa3
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A6: $o,B5: $o] :
                  ( X
                  = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ? [N2: nat] :
                    ( Xa3
                    = ( suc @ ( suc @ N2 ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ( Y
                         != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                            @ ( if_nat
                              @ ( ( ord_less_nat @ Xa3 @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa3 ) )
                              @ one_one_nat
                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                @ ( if_nat
                                  @ ( ( Xa3 = Mi2 )
                                    & ( Xa3 = Ma2 ) )
                                  @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                                    @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                          @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                @ ( if_nat
                                                  @ ( ( ( Xa3 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa3 != Mi2 )
                                                     => ( Xa3 = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                    @ ( plus_plus_nat @ one_one_nat
                                                      @ ( if_nat
                                                        @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                          = none_nat )
                                                        @ one_one_nat
                                                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ one_one_nat ) ) )
                                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                              @ ( if_nat
                                                @ ( ( ( Xa3 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa3 != Mi2 )
                                                   => ( Xa3 = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                @ one_one_nat ) ) ) ) )
                                      @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
thf(fact_5491_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa3 )
     => ( ! [Uu: $o,Uv: $o] :
            ( X
           != ( vEBT_Leaf @ Uu @ Uv ) )
       => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb ) )
               => ( ( Xa3 = Mi2 )
                  | ( Xa3 = 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 ) )
                 => ( ( Xa3 = Mi2 )
                    | ( Xa3 = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( 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 @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_5492_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa3 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => Y )
       => ( ( ? [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
           => Y )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb ) )
               => ( Y
                  = ( ~ ( ( Xa3 = Mi2 )
                        | ( Xa3 = 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
                    = ( ~ ( ( Xa3 = Mi2 )
                          | ( Xa3 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( 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 @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_5493_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa3 )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
           => ( ( ( Xa3 = zero_zero_nat )
               => A6 )
              & ( ( Xa3 != zero_zero_nat )
               => ( ( ( Xa3 = one_one_nat )
                   => B5 )
                  & ( Xa3 = one_one_nat ) ) ) ) )
       => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_5494_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_5495_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).

% buildup_nothing_in_leaf
thf(fact_5496_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X2: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X2 )
          | ( vEBT_VEBT_membermima @ T2 @ X2 ) ) ) ) ).

% both_member_options_def
thf(fact_5497_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_5498_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( set_or8404916559141939852nteger @ L @ ( plus_p5714425477246183910nteger @ U @ one_one_Code_integer ) )
      = ( set_or189985376899183464nteger @ L @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_5499_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_5500_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_5501_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) ).

% VEBT_internal.membermima.simps(2)
thf(fact_5502_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Leaf @ A @ B ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
thf(fact_5503_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_5504_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_5505_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ X )
      = ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_5506_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_a_x_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% maxt_bound
thf(fact_5507_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
thf(fact_5508_mint__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% mint_bound
thf(fact_5509_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Leaf @ A @ B ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
thf(fact_5510_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
thf(fact_5511_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_5512_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
thf(fact_5513_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% insert'_bound_height
thf(fact_5514_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X )
        = Y )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B5 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_5515_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X )
        = Y )
     => ( ! [A6: $o] :
            ( ? [B5: $o] :
                ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ A6 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_5516_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_5517_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList @ S ) @ 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_5518_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_5519_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ one_one_nat
          @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
            @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                @ ( if_nat
                  @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                     != none_nat )
                    & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                    @ ( if_nat
                      @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                        = none_nat )
                      @ ( plus_plus_nat @ one_one_nat @ one_one_nat )
                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
              @ one_one_nat ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
thf(fact_5520_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ one_one_nat
          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
            @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                  @ ( if_nat
                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                       != none_nat )
                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                      @ ( if_nat
                        @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          = none_nat )
                        @ one_one_nat
                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ one_one_nat ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
thf(fact_5521_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X @ Xa3 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( ( Xa3 = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A6: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ A6 @ Uw ) )
           => ( ( Xa3
                = ( suc @ zero_zero_nat ) )
             => ( Y
               != ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
         => ( ! [A6: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A6 @ B5 ) )
               => ( ? [Va: nat] :
                      ( Xa3
                      = ( suc @ ( suc @ Va ) ) )
                 => ( Y
                   != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B5 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ) )
           => ( ( ? [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va3 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ( Y
                         != ( plus_plus_nat @ one_one_nat
                            @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa3 ) @ one_one_nat
                              @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
                                @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                            = none_nat )
                                          @ ( plus_plus_nat @ one_one_nat @ one_one_nat )
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                  @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
thf(fact_5522_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X @ Xa3 )
        = Y )
     => ( ( ? [Uu: $o,B5: $o] :
              ( X
              = ( vEBT_Leaf @ Uu @ B5 ) )
         => ( ( Xa3 = zero_zero_nat )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ Uv @ Uw ) )
           => ( ? [N2: nat] :
                  ( Xa3
                  = ( suc @ N2 ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) )
             => ( Y != one_one_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 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( Y
                       != ( plus_plus_nat @ one_one_nat
                          @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ one_one_nat
                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
                              @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                            = none_nat )
                                          @ one_one_nat
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
thf(fact_5523_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa3 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb ) )
           => ~ ( ( Xa3 = Mi2 )
                | ( Xa3 = 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 ) )
             => ~ ( ( Xa3 = Mi2 )
                  | ( Xa3 = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( 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 @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_5524_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
        @ ( if_nat
          @ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
          @ one_one_nat
          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
            @ ( if_nat
              @ ( ( X = Mi )
                & ( X = Ma ) )
              @ ( numeral_numeral_nat @ ( bit1 @ one ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_d_e_l_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                            @ ( if_nat
                              @ ( ( ( X = Mi )
                                 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                    = Ma ) )
                                & ( ( X != Mi )
                                 => ( X = Ma ) ) )
                              @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                @ ( plus_plus_nat @ one_one_nat
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      = none_nat )
                                    @ one_one_nat
                                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                              @ one_one_nat ) ) )
                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                          @ ( if_nat
                            @ ( ( ( X = Mi )
                               => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                  = Ma ) )
                              & ( ( X != Mi )
                               => ( X = Ma ) ) )
                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                            @ one_one_nat ) ) ) ) )
                  @ one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
thf(fact_5525_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa3 )
        = Y )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
           => ( Y
              = ( ~ ( ( ( Xa3 = zero_zero_nat )
                     => A6 )
                    & ( ( Xa3 != zero_zero_nat )
                     => ( ( ( Xa3 = one_one_nat )
                         => B5 )
                        & ( Xa3 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
           => Y )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_5526_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa3 )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
           => ~ ( ( ( Xa3 = zero_zero_nat )
                 => A6 )
                & ( ( Xa3 != zero_zero_nat )
                 => ( ( ( Xa3 = one_one_nat )
                     => B5 )
                    & ( Xa3 = one_one_nat ) ) ) ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_5527_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Xa3 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A6: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A6 @ B5 ) )
               => ( ( Xa3
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A6: $o,B5: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A6 @ B5 ) )
                 => ! [N2: nat] :
                      ( ( Xa3
                        = ( suc @ ( suc @ N2 ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( Y
                              = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                @ ( if_nat
                                  @ ( ( ord_less_nat @ Xa3 @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa3 ) )
                                  @ one_one_nat
                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                    @ ( if_nat
                                      @ ( ( Xa3 = Mi2 )
                                        & ( Xa3 = Ma2 ) )
                                      @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                                        @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                              @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                    @ ( if_nat
                                                      @ ( ( ( Xa3 = Mi2 )
                                                         => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                            = Ma2 ) )
                                                        & ( ( Xa3 != Mi2 )
                                                         => ( Xa3 = Ma2 ) ) )
                                                      @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                        @ ( plus_plus_nat @ one_one_nat
                                                          @ ( if_nat
                                                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                              = none_nat )
                                                            @ one_one_nat
                                                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                                      @ one_one_nat ) ) )
                                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                  @ ( if_nat
                                                    @ ( ( ( Xa3 = Mi2 )
                                                       => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                          = Ma2 ) )
                                                      & ( ( Xa3 != Mi2 )
                                                       => ( Xa3 = Ma2 ) ) )
                                                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                    @ one_one_nat ) ) ) ) )
                                          @ one_one_nat ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_5528_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [Uu: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ B5 ) )
             => ( ( Xa3 = zero_zero_nat )
               => ( ( Y
                    = ( plus_plus_nat @ one_one_nat @ one_one_nat ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B5 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv @ Uw ) )
               => ! [N2: nat] :
                    ( ( Xa3
                      = ( suc @ N2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Xa3 ) ) ) )
             => ( ! [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 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa3 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa3 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( Y
                            = ( plus_plus_nat @ one_one_nat
                              @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ one_one_nat
                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
                                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                    @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                        @ ( if_nat
                                          @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                             != none_nat )
                                            & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                = none_nat )
                                              @ one_one_nat
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                    @ one_one_nat ) ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_5529_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( Xa3 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A6: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A6 @ Uw ) )
               => ( ( Xa3
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ Uw ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A6: $o,B5: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A6 @ B5 ) )
                 => ! [Va: nat] :
                      ( ( Xa3
                        = ( suc @ ( suc @ Va ) ) )
                     => ( ( Y
                          = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B5 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
             => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va3 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va3 ) @ Xa3 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa3 ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa3 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( Y
                              = ( plus_plus_nat @ one_one_nat
                                @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa3 ) @ one_one_nat
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
                                    @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                                        @ ( if_nat
                                          @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                             != none_nat )
                                            & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                = none_nat )
                                              @ ( plus_plus_nat @ one_one_nat @ one_one_nat )
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                      @ one_one_nat ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_5530_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y5: nat] :
                ( ( vEBT_vebt_member @ T @ Y5 )
                & ( ord_less_nat @ Y5 @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_5531_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y5: nat] :
                ( ( vEBT_vebt_member @ T @ Y5 )
                & ( ord_less_nat @ X @ Y5 ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_5532_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [Uu: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ B5 ) )
             => ( ( Xa3 = zero_zero_nat )
               => ( ( ( B5
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B5
                     => ( Y = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B5 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv @ Uw ) )
               => ! [N2: nat] :
                    ( ( Xa3
                      = ( suc @ N2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Xa3 ) ) ) )
             => ( ! [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 ) @ Xa3 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa3 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa3 @ Mi2 )
                             => ( Y
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( 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 @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ none_nat
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_5533_buildup__gives__empty,axiom,
    ! [N: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_5534_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_5535_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_5536_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_5537_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_5538_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_5539_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_5540_atLeastatMost__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( set_or189985376899183464nteger @ A @ B )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
thf(fact_5541_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_5542_atLeastLessThan__empty,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( set_or66887138388493659n_real @ A @ B )
        = bot_bot_set_real ) ) ).

% atLeastLessThan_empty
thf(fact_5543_atLeastLessThan__empty,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( set_or3540276404033026485et_nat @ A @ B )
        = bot_bot_set_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_5544_atLeastLessThan__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( set_or1222409239386451017an_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastLessThan_empty
thf(fact_5545_atLeastLessThan__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( set_or4665077453230672383an_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_5546_atLeastLessThan__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( set_or4662586982721622107an_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
thf(fact_5547_atLeastLessThan__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( set_or8404916559141939852nteger @ A @ B )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
thf(fact_5548_atLeastLessThan__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( set_or66887138388493659n_real @ A @ B )
        = bot_bot_set_real )
      = ( ~ ( ord_less_real @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5549_atLeastLessThan__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_rat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5550_atLeastLessThan__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_num @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5551_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_nat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5552_atLeastLessThan__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_int @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5553_atLeastLessThan__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = bot_bo3990330152332043303nteger )
      = ( ~ ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5554_atLeastLessThan__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real
        = ( set_or66887138388493659n_real @ A @ B ) )
      = ( ~ ( ord_less_real @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5555_atLeastLessThan__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or4029947393144176647an_rat @ A @ B ) )
      = ( ~ ( ord_less_rat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5556_atLeastLessThan__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or1222409239386451017an_num @ A @ B ) )
      = ( ~ ( ord_less_num @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5557_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or4665077453230672383an_nat @ A @ B ) )
      = ( ~ ( ord_less_nat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5558_atLeastLessThan__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or4662586982721622107an_int @ A @ B ) )
      = ( ~ ( ord_less_int @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5559_atLeastLessThan__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger
        = ( set_or8404916559141939852nteger @ A @ B ) )
      = ( ~ ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5560_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_5561_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_5562_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( Xa3 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A6: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A6 @ Uw ) )
               => ( ( Xa3
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A6
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A6
                       => ( Y = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ Uw ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A6: $o,B5: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A6 @ B5 ) )
                 => ! [Va: nat] :
                      ( ( Xa3
                        = ( suc @ ( suc @ Va ) ) )
                     => ( ( ( B5
                           => ( Y
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B5
                           => ( ( A6
                               => ( Y
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A6
                               => ( Y = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
             => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va3 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va3 ) @ Xa3 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa3 ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa3 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa3 )
                               => ( Y
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                               => ( Y
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( 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 @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                          = none_nat )
                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa3 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                    @ none_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_5563_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Xa3 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A6: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A6 @ B5 ) )
               => ( ( Xa3
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A6: $o,B5: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A6 @ B5 ) )
                 => ! [N2: nat] :
                      ( ( Xa3
                        = ( suc @ ( suc @ N2 ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa3 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa3 ) )
                               => ( Y = one_one_nat ) )
                              & ( ~ ( ( ord_less_nat @ Xa3 @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa3 ) )
                               => ( ( ( ( Xa3 = Mi2 )
                                      & ( Xa3 = Ma2 ) )
                                   => ( Y = one_one_nat ) )
                                  & ( ~ ( ( Xa3 = Mi2 )
                                        & ( Xa3 = Ma2 ) )
                                   => ( Y
                                      = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) @ one_one_nat ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
thf(fact_5564_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Xa3 = zero_zero_nat )
               => ( ( Y
                    = ( vEBT_Leaf @ $false @ B5 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A6: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A6 @ B5 ) )
               => ( ( Xa3
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( vEBT_Leaf @ A6 @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A6: $o,B5: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A6 @ B5 ) )
                 => ! [N2: nat] :
                      ( ( Xa3
                        = ( suc @ ( suc @ N2 ) ) )
                     => ( ( Y
                          = ( vEBT_Leaf @ A6 @ B5 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                     => ( ( Y
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ Xa3 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                       => ( ( Y
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ Xa3 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa3 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa3 ) )
                               => ( Y
                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa3 @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa3 ) )
                               => ( ( ( ( Xa3 = Mi2 )
                                      & ( Xa3 = Ma2 ) )
                                   => ( Y
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                                  & ( ~ ( ( Xa3 = Mi2 )
                                        & ( Xa3 = Ma2 ) )
                                   => ( Y
                                      = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                        @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa3 = Mi2 ) @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa3 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa3 != Mi2 )
                                                     => ( Xa3 = Ma2 ) ) )
                                                  @ ( if_nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      = none_nat )
                                                    @ ( if_nat @ ( Xa3 = Mi2 ) @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ Mi2 )
                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa3 = Mi2 ) @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa3 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa3 != Mi2 )
                                                     => ( Xa3 = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa3 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ Summary2 ) )
                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_5565_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [Uu: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ B5 ) )
             => ( ( Xa3 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B5 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv @ Uw ) )
               => ! [N2: nat] :
                    ( ( Xa3
                      = ( suc @ N2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Xa3 ) ) ) )
             => ( ! [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 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa3 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa3 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa3 @ Mi2 )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                             => ( Y
                                = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                  @ one_one_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_5566_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa3 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa3 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) @ Xa3 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ Xa3 ) ) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
                            @ ( if_nat
                              @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                & ~ ( ( Xa3 = Mi2 )
                                    | ( Xa3 = Ma2 ) ) )
                              @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                              @ one_one_nat ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_5567_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( ( ( Xa3 = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B5 ) ) )
                  & ( ( Xa3 != zero_zero_nat )
                   => ( ( ( Xa3 = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A6 @ $true ) ) )
                      & ( ( Xa3 != one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A6 @ B5 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa3 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
               => ( ( Y
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) @ Xa3 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ Xa3 ) ) ) )
             => ( ! [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 @ Xa3 @ Xa3 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                              & ~ ( ( Xa3 = Mi2 )
                                  | ( Xa3 = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Xa3 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_5568_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( Xa3 = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A6: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A6 @ Uw ) )
               => ( ( Xa3
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ Uw ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A6: $o,B5: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A6 @ B5 ) )
                 => ! [Va: nat] :
                      ( ( Xa3
                        = ( suc @ ( suc @ Va ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
             => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va3 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va3 ) @ Xa3 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa3 ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa3 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa3 )
                               => ( Y = one_one_nat ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                               => ( Y
                                  = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                    @ one_one_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_5569_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa3 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) @ Xa3 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ Xa3 ) ) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( if_nat
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                              & ~ ( ( Xa3 = Mi2 )
                                  | ( Xa3 = Ma2 ) ) )
                            @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ Mi2 @ Xa3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                            @ one_one_nat ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_5570_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Y
                  = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa3 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa3 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y
                    = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa3 ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
                 => ( ( Y
                      = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) @ Xa3 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) )
                   => ( ( Y
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) @ Xa3 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa3 = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa3 = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa3 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_5571_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa3 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa3 ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) @ Xa3 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) @ Xa3 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ one_one_nat
                            @ ( if_nat @ ( Xa3 = Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( Xa3 = Ma2 ) @ zero_zero_nat
                                @ ( if_nat @ ( ord_less_nat @ Xa3 @ Mi2 ) @ zero_zero_nat
                                  @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa3 ) @ zero_zero_nat
                                    @ ( if_nat
                                      @ ( ( ord_less_nat @ Mi2 @ Xa3 )
                                        & ( ord_less_nat @ Xa3 @ Ma2 ) )
                                      @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                                      @ zero_zero_nat ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_5572_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_real,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( size_size_list_real @ TreeList4 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).

% tcd
thf(fact_5573_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_o,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( size_size_list_o @ TreeList4 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).

% tcd
thf(fact_5574_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_nat,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( size_size_list_nat @ TreeList4 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).

% tcd
thf(fact_5575_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList4: list_int,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( size_size_list_int @ TreeList4 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).

% tcd
thf(fact_5576_txe,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ Y @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y ) @ ( nth_VEBT_VEBTi @ Tree_is @ Y ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).

% txe
thf(fact_5577_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X ) )
        = ( minus_minus_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% delete_correct'
thf(fact_5578_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X ) )
        = ( minus_minus_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% delete_correct
thf(fact_5579_local_Oext,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ Y @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y ) @ ( nth_VEBT_VEBTi @ Tree_is @ Y ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y ) @ ( nth_VEBT_VEBTi @ Tree_is @ Y ) ) ) ) ) ).

% local.ext
thf(fact_5580_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_5581_recomp,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ I ) @ ( nth_VEBT_VEBTi @ Tree_is @ I ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).

% recomp
thf(fact_5582_repack,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,Rest: assn,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ I ) @ ( nth_VEBT_VEBTi @ Tree_is @ I ) ) @ Rest ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ Rest @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).

% repack
thf(fact_5583_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_5584_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_5585_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( set_or1269000886237332187st_nat @ M @ N )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_5586_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_5587_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% atLeastAtMost_insertL
thf(fact_5588_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ ( insert_real @ X @ ( set_real2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5589_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ ( insert_int @ X @ ( set_int2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5590_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5591_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] : ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) ) @ ( insert_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5592_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ ( insert_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_5593_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or66887138388493659n_real
    = ( ^ [A4: real,B2: real] : ( minus_minus_set_real @ ( set_or1222579329274155063t_real @ A4 @ B2 ) @ ( insert_real @ B2 @ bot_bot_set_real ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_5594_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [A4: nat,B2: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A4 @ B2 ) @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_5595_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [A4: int,B2: int] : ( minus_minus_set_int @ ( set_or1266510415728281911st_int @ A4 @ B2 ) @ ( insert_int @ B2 @ bot_bot_set_int ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_5596_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or8404916559141939852nteger
    = ( ^ [A4: code_integer,B2: code_integer] : ( minus_2355218937544613996nteger @ ( set_or189985376899183464nteger @ A4 @ B2 ) @ ( insert_Code_integer @ B2 @ bot_bo3990330152332043303nteger ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_5597_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_5598_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
     => ( ( insert_VEBT_VEBT @ ( nth_VEBT_VEBT @ L @ I ) @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L @ I @ X ) ) )
        = ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5599_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L ) )
     => ( ( insert_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ L @ I ) @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L @ I @ X ) ) )
        = ( insert_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5600_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_real,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L ) )
     => ( ( insert_real @ ( nth_real @ L @ I ) @ ( set_real2 @ ( list_update_real @ L @ I @ X ) ) )
        = ( insert_real @ X @ ( set_real2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5601_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L ) )
     => ( ( insert_o @ ( nth_o @ L @ I ) @ ( set_o2 @ ( list_update_o @ L @ I @ X ) ) )
        = ( insert_o @ X @ ( set_o2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5602_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( insert_nat @ ( nth_nat @ L @ I ) @ ( set_nat2 @ ( list_update_nat @ L @ I @ X ) ) )
        = ( insert_nat @ X @ ( set_nat2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5603_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( insert_int @ ( nth_int @ L @ I ) @ ( set_int2 @ ( list_update_int @ L @ I @ X ) ) )
        = ( insert_int @ X @ ( set_int2 @ L ) ) ) ) ).

% insert_swap_set_eq
thf(fact_5604_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: vEBT_VEBT > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ ( insert_nat @ I @ I5 ) @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5605_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: vEBT_VEBT > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L8650695023172932196BT_nat @ ( insert_nat @ I @ I5 ) @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5606_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: vEBT_VEBT > int > assn,Xsi: list_int] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L8648204552663881920BT_int @ ( insert_nat @ I @ I5 ) @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8648204552663881920BT_int @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5607_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: vEBT_VEBTi > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ ( insert_nat @ I @ I5 ) @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5608_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: vEBT_VEBTi > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ ( insert_nat @ I @ I5 ) @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5609_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: vEBT_VEBTi > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2809031099982602151Ti_nat @ ( insert_nat @ I @ I5 ) @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5610_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: vEBT_VEBTi > int > assn,Xsi: list_int] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2806540629473551875Ti_int @ ( insert_nat @ I @ I5 ) @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5611_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: real > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ ( insert_nat @ I @ I5 ) @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5612_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: real > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ ( insert_nat @ I @ I5 ) @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5613_listI__assn__insert,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: real > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L234762979517870878al_nat @ ( insert_nat @ I @ I5 ) @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_5614_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: vEBT_VEBT > vEBT_VEBT > assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ ( insert_nat @ I @ I5 ) @ A2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A2 @ X1 @ X22 ) @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5615_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: vEBT_VEBTi > vEBT_VEBT > assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ ( insert_nat @ I @ I5 ) @ A2 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A2 @ X1 @ X22 ) @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5616_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: vEBT_VEBTi > vEBT_VEBTi > assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ ( insert_nat @ I @ I5 ) @ A2 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A2 @ X1 @ X22 ) @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5617_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: real > vEBT_VEBT > assn,X1: real,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ ( insert_nat @ I @ I5 ) @ A2 @ ( list_update_real @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A2 @ X1 @ X22 ) @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5618_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: real > vEBT_VEBTi > assn,X1: real,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ ( insert_nat @ I @ I5 ) @ A2 @ ( list_update_real @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A2 @ X1 @ X22 ) @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5619_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_o,A2: $o > vEBT_VEBT > assn,X1: $o,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( vEBT_L1319876754960170684T_VEBT @ ( insert_nat @ I @ I5 ) @ A2 @ ( list_update_o @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A2 @ X1 @ X22 ) @ ( vEBT_L1319876754960170684T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5620_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_o,A2: $o > vEBT_VEBTi > assn,X1: $o,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( vEBT_L6286945158656146733_VEBTi @ ( insert_nat @ I @ I5 ) @ A2 @ ( list_update_o @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A2 @ X1 @ X22 ) @ ( vEBT_L6286945158656146733_VEBTi @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5621_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A2: nat > vEBT_VEBT > assn,X1: nat,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( vEBT_L8511957252848910786T_VEBT @ ( insert_nat @ I @ I5 ) @ A2 @ ( list_update_nat @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A2 @ X1 @ X22 ) @ ( vEBT_L8511957252848910786T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5622_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_nat,A2: nat > vEBT_VEBTi > assn,X1: nat,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( vEBT_L7489483478785760935_VEBTi @ ( insert_nat @ I @ I5 ) @ A2 @ ( list_update_nat @ Xs @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A2 @ X1 @ X22 ) @ ( vEBT_L7489483478785760935_VEBTi @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5623_listI__assn__subst,axiom,
    ! [I: nat,I5: set_nat,Xs: list_int,A2: int > vEBT_VEBT > assn,X1: int,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( vEBT_L2018189785592951398T_VEBT @ ( insert_nat @ I @ I5 ) @ A2 @ ( list_update_int @ Xs @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A2 @ X1 @ X22 ) @ ( vEBT_L2018189785592951398T_VEBT @ I5 @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_5624_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_5625_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: vEBT_VEBT > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ I5 @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5626_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: vEBT_VEBT > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L8650695023172932196BT_nat @ I5 @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5627_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBT,A2: vEBT_VEBT > int > assn,Xsi: list_int] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( vEBT_L8648204552663881920BT_int @ I5 @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8648204552663881920BT_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5628_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: vEBT_VEBTi > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ I5 @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5629_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: vEBT_VEBTi > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ I5 @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5630_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: vEBT_VEBTi > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2809031099982602151Ti_nat @ I5 @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5631_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_VEBT_VEBTi,A2: vEBT_VEBTi > int > assn,Xsi: list_int] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( vEBT_L2806540629473551875Ti_int @ I5 @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5632_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: real > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ I5 @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5633_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: real > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ I5 @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5634_listI__assn__extract,axiom,
    ! [I: nat,I5: set_nat,Xs: list_real,A2: real > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I5 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( vEBT_L234762979517870878al_nat @ I5 @ A2 @ Xs @ Xsi )
          = ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_5635_listI__assn__reinsert,axiom,
    ! [P: assn,A2: vEBT_VEBT > vEBT_VEBT > assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5636_listI__assn__reinsert,axiom,
    ! [P: assn,A2: vEBT_VEBT > nat > assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8650695023172932196BT_nat @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5637_listI__assn__reinsert,axiom,
    ! [P: assn,A2: vEBT_VEBT > int > assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_int,I5: set_nat,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8648204552663881920BT_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8648204552663881920BT_int @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5638_listI__assn__reinsert,axiom,
    ! [P: assn,A2: vEBT_VEBTi > vEBT_VEBT > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5639_listI__assn__reinsert,axiom,
    ! [P: assn,A2: vEBT_VEBTi > vEBT_VEBTi > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5640_listI__assn__reinsert,axiom,
    ! [P: assn,A2: vEBT_VEBTi > nat > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2809031099982602151Ti_nat @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5641_listI__assn__reinsert,axiom,
    ! [P: assn,A2: vEBT_VEBTi > int > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_int,I5: set_nat,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2806540629473551875Ti_int @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5642_listI__assn__reinsert,axiom,
    ! [P: assn,A2: real > vEBT_VEBT > assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5643_listI__assn__reinsert,axiom,
    ! [P: assn,A2: real > vEBT_VEBTi > assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5644_listI__assn__reinsert,axiom,
    ! [P: assn,A2: real > nat > assn,Xs: list_real,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L234762979517870878al_nat @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_5645_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: vEBT_VEBT > vEBT_VEBT > assn,X: vEBT_VEBT,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi2 ) ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5646_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: vEBT_VEBTi > vEBT_VEBT > assn,X: vEBT_VEBTi,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A2 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi2 ) ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5647_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: vEBT_VEBTi > vEBT_VEBTi > assn,X: vEBT_VEBTi,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A2 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi2 ) ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5648_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: real > vEBT_VEBT > assn,X: real,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A2 @ ( list_update_real @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi2 ) ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5649_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: real > vEBT_VEBTi > assn,X: real,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A2 @ ( list_update_real @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi2 ) ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5650_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: $o > vEBT_VEBT > assn,X: $o,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBT,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L1319876754960170684T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L1319876754960170684T_VEBT @ I5 @ A2 @ ( list_update_o @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi2 ) ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5651_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: $o > vEBT_VEBTi > assn,X: $o,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBTi,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L6286945158656146733_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L6286945158656146733_VEBTi @ I5 @ A2 @ ( list_update_o @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi2 ) ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5652_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: nat > vEBT_VEBT > assn,X: nat,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBT,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L8511957252848910786T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8511957252848910786T_VEBT @ I5 @ A2 @ ( list_update_nat @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi2 ) ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5653_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: nat > vEBT_VEBTi > assn,X: nat,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBTi,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L7489483478785760935_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7489483478785760935_VEBTi @ I5 @ A2 @ ( list_update_nat @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi2 ) ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5654_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A2: int > vEBT_VEBT > assn,X: int,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_int,Xsi: list_VEBT_VEBT,F2: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L2018189785592951398T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2018189785592951398T_VEBT @ I5 @ A2 @ ( list_update_int @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi2 ) ) @ F2 ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_5655_big__assn__simp,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,L: nat,X: vEBT_VEBTi,Xaa: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ H2 @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails
        @ ( times_times_assn
          @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) @ X )
            @ ( pure_assn
              @ ( Xaa
                = ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) ) ) )
          @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ H2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) )
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
            @ ( pure_assn
              @ ( Xaa
                = ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) ) ) )
          @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) ) ) ) ).

% big_assn_simp
thf(fact_5656_big__assn__simp_H,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,Xaa: vEBT_VEBT,L: nat,X: vEBT_VEBTi,Xb3: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ H2 @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( Xaa
          = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
       => ( entails
          @ ( times_times_assn
            @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Xaa @ X )
              @ ( pure_assn
                @ ( Xb3
                  = ( vEBT_vebt_mint @ Xaa ) ) ) )
            @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ H2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) )
          @ ( times_times_assn
            @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
              @ ( pure_assn
                @ ( Xb3
                  = ( vEBT_vebt_mint @ Xaa ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Xaa ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) ) ) ) ) ).

% big_assn_simp'
thf(fact_5657_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa3 ) )
               => ( ( ( Xa3 = zero_zero_nat )
                   => A6 )
                  & ( ( Xa3 != zero_zero_nat )
                   => ( ( ( Xa3 = one_one_nat )
                       => B5 )
                      & ( Xa3 = 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 ) @ Xa3 ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) @ Xa3 ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) @ Xa3 ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) )
                       => ( ( Xa3 != Mi2 )
                         => ( ( Xa3 != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_5658_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Y
                  = ( ( ( Xa3 = zero_zero_nat )
                     => A6 )
                    & ( ( Xa3 != zero_zero_nat )
                     => ( ( ( Xa3 = one_one_nat )
                         => B5 )
                        & ( Xa3 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa3 ) ) ) )
         => ( ! [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 ) @ Xa3 ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
                 => ( ~ Y
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) @ Xa3 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) )
                   => ( ~ Y
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc2 ) @ Xa3 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
                     => ( ( Y
                          = ( ( Xa3 != Mi2 )
                           => ( ( Xa3 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_5659_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa3 ) )
               => ( ( ( Xa3 = zero_zero_nat )
                   => A6 )
                  & ( ( Xa3 != zero_zero_nat )
                   => ( ( ( Xa3 = one_one_nat )
                       => B5 )
                      & ( Xa3 = 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 ) @ Xa3 ) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S3 ) @ Xa3 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_5660_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa3 ) )
               => ~ ( ( ( Xa3 = zero_zero_nat )
                     => A6 )
                    & ( ( Xa3 != zero_zero_nat )
                     => ( ( ( Xa3 = one_one_nat )
                         => B5 )
                        & ( Xa3 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S3 ) @ Xa3 ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_5661_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
     => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_5662_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I2: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I2 ) @ bot_bot_set_int @ ( insert_int @ I2 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_5663_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ( vEBT_vebt_member @ X @ Xa3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa3 ) )
               => ~ ( ( ( Xa3 = zero_zero_nat )
                     => A6 )
                    & ( ( Xa3 != zero_zero_nat )
                     => ( ( ( Xa3 = one_one_nat )
                         => B5 )
                        & ( Xa3 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa3 ) )
                 => ~ ( ( Xa3 != Mi2 )
                     => ( ( Xa3 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa3 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa3 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_5664_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Y
                  = ( ( ( Xa3 = zero_zero_nat )
                     => A6 )
                    & ( ( Xa3 != zero_zero_nat )
                     => ( ( ( Xa3 = one_one_nat )
                         => B5 )
                        & ( Xa3 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A6 @ B5 ) @ Xa3 ) ) ) )
         => ( ! [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 ) @ Xa3 ) ) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S3 ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList2 @ S3 ) @ Xa3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_5665_IH,axiom,
    ! [Xa3: nat,Xb3: nat,Ti: vEBT_VEBTi] :
      ( ( xa != mi )
     => ( ( xa != ma )
       => ( ~ ( ord_less_nat @ xa @ mi )
         => ( ~ ( ord_less_nat @ ma @ xa )
           => ( ( Xa3
                = ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
             => ( ( Xb3
                  = ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
               => ( ( ord_less_nat @ Xa3 @ ( size_s6755466524823107622T_VEBT @ treeList ) )
                 => ( hoare_hoare_triple_o @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ treeList @ Xa3 ) @ Ti ) @ ( vEBT_V854960066525838166emberi @ ( nth_VEBT_VEBT @ treeList @ Xa3 ) @ Ti @ Xb3 )
                    @ ^ [R5: $o] :
                        ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ treeList @ Xa3 ) @ Ti )
                        @ ( pure_assn
                          @ ( R5
                            = ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ Xa3 ) @ Xb3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% IH
thf(fact_5666_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa3 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa3 ) ) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Xa3 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb ) )
                 => ( ( Y
                      = ( ( Xa3 = Mi2 )
                        | ( Xa3 = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb ) @ Xa3 ) ) ) )
             => ( ! [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
                        = ( ( Xa3 = Mi2 )
                          | ( Xa3 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( 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 ) @ Xa3 ) ) ) )
               => ~ ! [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 @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( 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 ) @ Xa3 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_5667_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa3 ) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Xa3 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb ) @ Xa3 ) )
                   => ( ( Xa3 = Mi2 )
                      | ( Xa3 = 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 ) @ Xa3 ) )
                     => ( ( Xa3 = Mi2 )
                        | ( Xa3 = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( 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 ) @ Xa3 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_5668_ent__pure__pre__iff,axiom,
    ! [P: assn,B: $o,Q: assn] :
      ( ( entails @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ Q )
      = ( B
       => ( entails @ P @ Q ) ) ) ).

% ent_pure_pre_iff
thf(fact_5669_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa3: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa3 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb ) @ Xa3 ) )
               => ~ ( ( Xa3 = Mi2 )
                    | ( Xa3 = 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 ) @ Xa3 ) )
                 => ~ ( ( Xa3 = Mi2 )
                      | ( Xa3 = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( 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 ) @ Xa3 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_5670_heaphelp,axiom,
    ! [Xa3: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb3: vEBT_VEBTi,N: nat,Xc: vEBT_VEBTi,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Xa3 @ Tree_is ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb3 ) )
            @ ( pure_assn
              @ ( ( none_nat = none_nat )
                & ( N = N ) ) ) )
          @ ( pure_assn
            @ ( Xc
              = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ N @ Xa3 @ Xb3 ) ) ) )
        @ H2 )
     => ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ none_P5556105721700978146at_nat @ N @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).

% heaphelp
thf(fact_5671_heaphelp,axiom,
    ! [Xa3: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb3: vEBT_VEBTi,N: nat,Xc: vEBT_VEBTi,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Xa3 @ Tree_is ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb3 ) )
            @ ( pure_assn
              @ ( ( none_P5556105721700978146at_nat = none_P5556105721700978146at_nat )
                & ( N = N ) ) ) )
          @ ( pure_assn
            @ ( Xc
              = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ N @ Xa3 @ Xb3 ) ) ) )
        @ H2 )
     => ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ none_P5556105721700978146at_nat @ N @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).

% heaphelp
thf(fact_5672_heaphelp,axiom,
    ! [Xa3: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb3: vEBT_VEBTi,N: nat,Xc: vEBT_VEBTi,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Xa3 @ Tree_is ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb3 ) )
            @ ( pure_assn
              @ ( ( none_num = none_num )
                & ( N = N ) ) ) )
          @ ( pure_assn
            @ ( Xc
              = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ N @ Xa3 @ Xb3 ) ) ) )
        @ H2 )
     => ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ none_P5556105721700978146at_nat @ N @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).

% heaphelp
thf(fact_5673_merge__pure__star,axiom,
    ! [A: $o,B: $o] :
      ( ( times_times_assn @ ( pure_assn @ A ) @ ( pure_assn @ B ) )
      = ( pure_assn
        @ ( A
          & B ) ) ) ).

% merge_pure_star
thf(fact_5674_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Q: assn] :
      ( ( entails @ ( pure_assn @ B ) @ Q )
      = ( B
       => ( entails @ one_one_assn @ Q ) ) ) ).

% ent_pure_pre_iff_sng
thf(fact_5675_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(5)
thf(fact_5676_mod__pure__star__dist,axiom,
    ! [P: assn,B: $o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ H2 )
      = ( ( rep_assn @ P @ H2 )
        & B ) ) ).

% mod_pure_star_dist
thf(fact_5677_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B: $o] :
      ( ( entails @ P @ ( pure_assn @ B ) )
      = ( ! [H: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H )
           => B )
        & ( entails @ P @ one_one_assn ) ) ) ).

% ent_pure_post_iff_sng
thf(fact_5678_mod__h__bot__iff_I4_J,axiom,
    ! [Q2: array_VEBT_VEBTi,Y: list_VEBT_VEBTi,H2: heap_e7401611519738050253t_unit] :
      ~ ( rep_assn @ ( snga_assn_VEBT_VEBTi @ Q2 @ Y ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_h_bot_iff(4)
thf(fact_5679_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B: $o] :
      ( ( entails @ P @ ( times_times_assn @ Q @ ( pure_assn @ B ) ) )
      = ( ! [H: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H )
           => B )
        & ( entails @ P @ Q ) ) ) ).

% ent_pure_post_iff
thf(fact_5680_mod__starD,axiom,
    ! [A2: assn,B3: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A2 @ B3 ) @ H2 )
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( ( rep_assn @ A2 @ H1 )
          & ( rep_assn @ B3 @ H22 ) ) ) ).

% mod_starD
thf(fact_5681_mod__starE,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A @ B ) @ H2 )
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : ( rep_assn @ A @ X_1 )
         => ! [H_2: produc3658429121746597890et_nat] :
              ~ ( rep_assn @ B @ H_2 ) ) ) ).

% mod_starE
thf(fact_5682_entails__def,axiom,
    ( entails
    = ( ^ [P6: assn,Q7: assn] :
        ! [H: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P6 @ H )
         => ( rep_assn @ Q7 @ H ) ) ) ) ).

% entails_def
thf(fact_5683_entailsI,axiom,
    ! [P: assn,Q: assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H3 )
         => ( rep_assn @ Q @ H3 ) )
     => ( entails @ P @ Q ) ) ).

% entailsI
thf(fact_5684_entailsD,axiom,
    ! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
      ( ( entails @ P @ Q )
     => ( ( rep_assn @ P @ H2 )
       => ( rep_assn @ Q @ H2 ) ) ) ).

% entailsD
thf(fact_5685_ent__fwd,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat,Q: assn] :
      ( ( rep_assn @ P @ H2 )
     => ( ( entails @ P @ Q )
       => ( rep_assn @ Q @ H2 ) ) ) ).

% ent_fwd
thf(fact_5686_extract__pre__list__assn__lengthD,axiom,
    ! [A2: real > real > assn,Xs: list_real,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L1930518968523514909l_real @ A2 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_real @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5687_extract__pre__list__assn__lengthD,axiom,
    ! [A2: $o > real > assn,Xs: list_o,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L4725278957065240257o_real @ A2 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_o @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5688_extract__pre__list__assn__lengthD,axiom,
    ! [A2: nat > real > assn,Xs: list_nat,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L6102073776069194049t_real @ A2 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_nat @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5689_extract__pre__list__assn__lengthD,axiom,
    ! [A2: int > real > assn,Xs: list_int,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L8288995350762215837t_real @ A2 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_int @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5690_extract__pre__list__assn__lengthD,axiom,
    ! [A2: real > $o > assn,Xs: list_real,Xsi: list_o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L6234343332106409831real_o @ A2 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_o @ Xsi )
        = ( size_size_list_real @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5691_extract__pre__list__assn__lengthD,axiom,
    ! [A2: $o > $o > assn,Xs: list_o,Xsi: list_o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L7363604446928714179sn_o_o @ A2 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_o @ Xsi )
        = ( size_size_list_o @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5692_extract__pre__list__assn__lengthD,axiom,
    ! [A2: nat > $o > assn,Xs: list_nat,Xsi: list_o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L7887682484454631235_nat_o @ A2 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_o @ Xsi )
        = ( size_size_list_nat @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5693_extract__pre__list__assn__lengthD,axiom,
    ! [A2: int > $o > assn,Xs: list_int,Xsi: list_o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L6066640139021943271_int_o @ A2 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_o @ Xsi )
        = ( size_size_list_int @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5694_extract__pre__list__assn__lengthD,axiom,
    ! [A2: real > nat > assn,Xs: list_real,Xsi: list_nat,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L1446010312343316929al_nat @ A2 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_nat @ Xsi )
        = ( size_size_list_real @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5695_extract__pre__list__assn__lengthD,axiom,
    ! [A2: $o > nat > assn,Xs: list_o,Xsi: list_nat,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L4785011123346445925_o_nat @ A2 @ Xs @ Xsi ) @ H2 )
     => ( ( size_size_list_nat @ Xsi )
        = ( size_size_list_o @ Xs ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_5696_assn__one__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ one_one_assn @ P )
      = P ) ).

% assn_one_left
thf(fact_5697_assn__times__comm,axiom,
    ( times_times_assn
    = ( ^ [P6: assn,Q7: assn] : ( times_times_assn @ Q7 @ P6 ) ) ) ).

% assn_times_comm
thf(fact_5698_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R3: assn] :
      ( ( times_times_assn @ ( times_times_assn @ P @ Q ) @ R3 )
      = ( times_times_assn @ P @ ( times_times_assn @ Q @ R3 ) ) ) ).

% assn_times_assoc
thf(fact_5699_ent__iffI,axiom,
    ! [A2: assn,B3: assn] :
      ( ( entails @ A2 @ B3 )
     => ( ( entails @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% ent_iffI
thf(fact_5700_ent__refl,axiom,
    ! [P: assn] : ( entails @ P @ P ) ).

% ent_refl
thf(fact_5701_ent__trans,axiom,
    ! [P: assn,Q: assn,R3: assn] :
      ( ( entails @ P @ Q )
     => ( ( entails @ Q @ R3 )
       => ( entails @ P @ R3 ) ) ) ).

% ent_trans
thf(fact_5702_ent__star__mono,axiom,
    ! [P: assn,P2: assn,Q: assn,Q5: assn] :
      ( ( entails @ P @ P2 )
     => ( ( entails @ Q @ Q5 )
       => ( entails @ ( times_times_assn @ P @ Q ) @ ( times_times_assn @ P2 @ Q5 ) ) ) ) ).

% ent_star_mono
thf(fact_5703_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: vEBT_VEBT > vEBT_VEBT > assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5704_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: vEBT_VEBT > nat > assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L8650695023172932196BT_nat @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5705_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: vEBT_VEBT > int > assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_int,I5: set_nat,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8648204552663881920BT_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L8648204552663881920BT_int @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5706_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: vEBT_VEBTi > vEBT_VEBT > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5707_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: vEBT_VEBTi > vEBT_VEBTi > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5708_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: vEBT_VEBTi > nat > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L2809031099982602151Ti_nat @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5709_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: vEBT_VEBTi > int > assn,Xs: list_VEBT_VEBTi,I: nat,Xsi: list_int,I5: set_nat,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L2806540629473551875Ti_int @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5710_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: real > vEBT_VEBT > assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBT,I5: set_nat,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5711_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: real > vEBT_VEBTi > assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBTi,I5: set_nat,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5712_listI__assn__reinsert_H,axiom,
    ! [P: assn,A2: real > nat > assn,Xs: list_real,I: nat,Xsi: list_nat,I5: set_nat,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L234762979517870878al_nat @ I5 @ A2 @ Xs @ Xsi ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_5713_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: vEBT_VEBT > vEBT_VEBT > assn,X: vEBT_VEBT,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I5 @ A2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi2 ) ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5714_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: vEBT_VEBTi > vEBT_VEBT > assn,X: vEBT_VEBTi,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I5 @ A2 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi2 ) ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5715_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: vEBT_VEBTi > vEBT_VEBTi > assn,X: vEBT_VEBTi,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I5 @ A2 @ ( list_u6098035379799741383_VEBTi @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi2 ) ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5716_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: real > vEBT_VEBT > assn,X: real,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I5 @ A2 @ ( list_update_real @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi2 ) ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5717_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: real > vEBT_VEBTi > assn,X: real,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I5 @ A2 @ ( list_update_real @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi2 ) ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5718_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: $o > vEBT_VEBT > assn,X: $o,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBT,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L1319876754960170684T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L1319876754960170684T_VEBT @ I5 @ A2 @ ( list_update_o @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi2 ) ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5719_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: $o > vEBT_VEBTi > assn,X: $o,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBTi,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L6286945158656146733_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L6286945158656146733_VEBTi @ I5 @ A2 @ ( list_update_o @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi2 ) ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5720_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: nat > vEBT_VEBT > assn,X: nat,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBT,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L8511957252848910786T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L8511957252848910786T_VEBT @ I5 @ A2 @ ( list_update_nat @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi2 ) ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5721_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: nat > vEBT_VEBTi > assn,X: nat,Xi2: vEBT_VEBTi,I5: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBTi,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L7489483478785760935_VEBTi @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L7489483478785760935_VEBTi @ I5 @ A2 @ ( list_update_nat @ Xs @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi2 ) ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5722_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A2: int > vEBT_VEBT > assn,X: int,Xi2: vEBT_VEBT,I5: set_nat,I: nat,Xs: list_int,Xsi: list_VEBT_VEBT,F2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A2 @ X @ Xi2 ) @ ( vEBT_L2018189785592951398T_VEBT @ ( minus_minus_set_nat @ I5 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L2018189785592951398T_VEBT @ I5 @ A2 @ ( list_update_int @ Xs @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi2 ) ) @ F2 ) @ C @ Q )
           => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_5723_rule__at__index,axiom,
    ! [P: assn,A2: vEBT_VEBT > vEBT_VEBT > assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F2: assn,I: nat,C: heap_Time_Heap_o,Q5: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L1279224858307276611T_VEBT @ A2 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) @ C @ Q5 )
         => ( ! [R: $o] : ( entails @ ( Q5 @ R ) @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ ( F4 @ R ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L1279224858307276611T_VEBT @ A2 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5724_rule__at__index,axiom,
    ! [P: assn,A2: vEBT_VEBT > nat > assn,Xs: list_VEBT_VEBT,Xsi: list_nat,F2: assn,I: nat,C: heap_Time_Heap_o,Q5: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L8296926524756676353BT_nat @ A2 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) @ C @ Q5 )
         => ( ! [R: $o] : ( entails @ ( Q5 @ R ) @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ ( F4 @ R ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L8296926524756676353BT_nat @ A2 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5725_rule__at__index,axiom,
    ! [P: assn,A2: vEBT_VEBT > int > assn,Xs: list_VEBT_VEBT,Xsi: list_int,F2: assn,I: nat,C: heap_Time_Heap_o,Q5: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L8294436054247626077BT_int @ A2 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8648204552663881920BT_int @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) @ C @ Q5 )
         => ( ! [R: $o] : ( entails @ ( Q5 @ R ) @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L8648204552663881920BT_int @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ ( F4 @ R ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L8294436054247626077BT_int @ A2 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5726_rule__at__index,axiom,
    ! [P: assn,A2: vEBT_VEBTi > vEBT_VEBT > assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F2: assn,I: nat,C: heap_Time_Heap_o,Q5: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L7265847600308530106T_VEBT @ A2 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) @ C @ Q5 )
         => ( ! [R: $o] : ( entails @ ( Q5 @ R ) @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ ( F4 @ R ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L7265847600308530106T_VEBT @ A2 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5727_rule__at__index,axiom,
    ! [P: assn,A2: vEBT_VEBTi > vEBT_VEBTi > assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F2: assn,I: nat,C: heap_Time_Heap_o,Q5: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L1891944875198410415_VEBTi @ A2 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) @ C @ Q5 )
         => ( ! [R: $o] : ( entails @ ( Q5 @ R ) @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ ( F4 @ R ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L1891944875198410415_VEBTi @ A2 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5728_rule__at__index,axiom,
    ! [P: assn,A2: vEBT_VEBTi > nat > assn,Xs: list_VEBT_VEBTi,Xsi: list_nat,F2: assn,I: nat,C: heap_Time_Heap_o,Q5: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L8930081998596925642Ti_nat @ A2 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) @ C @ Q5 )
         => ( ! [R: $o] : ( entails @ ( Q5 @ R ) @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2809031099982602151Ti_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ ( F4 @ R ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L8930081998596925642Ti_nat @ A2 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5729_rule__at__index,axiom,
    ! [P: assn,A2: vEBT_VEBTi > int > assn,Xs: list_VEBT_VEBTi,Xsi: list_int,F2: assn,I: nat,C: heap_Time_Heap_o,Q5: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L8927591528087875366Ti_int @ A2 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) @ C @ Q5 )
         => ( ! [R: $o] : ( entails @ ( Q5 @ R ) @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_VEBT_VEBTi @ Xs @ I ) @ ( nth_int @ Xsi @ I ) ) @ ( vEBT_L2806540629473551875Ti_int @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ ( F4 @ R ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L8927591528087875366Ti_int @ A2 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5730_rule__at__index,axiom,
    ! [P: assn,A2: real > vEBT_VEBT > assn,Xs: list_real,Xsi: list_VEBT_VEBT,F2: assn,I: nat,C: heap_Time_Heap_o,Q5: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L4595930785310033027T_VEBT @ A2 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) @ C @ Q5 )
         => ( ! [R: $o] : ( entails @ ( Q5 @ R ) @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ ( F4 @ R ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L4595930785310033027T_VEBT @ A2 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5731_rule__at__index,axiom,
    ! [P: assn,A2: real > vEBT_VEBTi > assn,Xs: list_real,Xsi: list_VEBT_VEBTi,F2: assn,I: nat,C: heap_Time_Heap_o,Q5: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L9060850011106065574_VEBTi @ A2 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) @ C @ Q5 )
         => ( ! [R: $o] : ( entails @ ( Q5 @ R ) @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ ( F4 @ R ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L9060850011106065574_VEBTi @ A2 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5732_rule__at__index,axiom,
    ! [P: assn,A2: real > nat > assn,Xs: list_real,Xsi: list_nat,F2: assn,I: nat,C: heap_Time_Heap_o,Q5: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L1446010312343316929al_nat @ A2 @ Xs @ Xsi ) @ F2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ F2 ) @ C @ Q5 )
         => ( ! [R: $o] : ( entails @ ( Q5 @ R ) @ ( times_times_assn @ ( times_times_assn @ ( A2 @ ( nth_real @ Xs @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A2 @ Xs @ Xsi ) ) @ ( F4 @ R ) ) )
           => ( hoare_hoare_triple_o @ P @ C
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L1446010312343316929al_nat @ A2 @ Xs @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5733_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_hoare_triple_o @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q )
      = ( B
       => ( hoare_hoare_triple_o @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_5734_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( hoare_7629718768684598413on_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q )
      = ( B
       => ( hoare_7629718768684598413on_nat @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_5735_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q )
      = ( B
       => ( hoare_3067605981109127869le_nat @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_5736_norm__pre__pure__iff,axiom,
    ! [P: assn,B: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q )
      = ( B
       => ( hoare_1429296392585015714_VEBTi @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_5737_foldr__zero,axiom,
    ! [Xs: list_nat,D2: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
         => ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ Xs @ I3 ) ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( minus_minus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs @ D2 ) @ D2 ) ) ) ).

% foldr_zero
thf(fact_5738_foldr__same,axiom,
    ! [Xs: list_real,Y: real] :
      ( ! [X3: real,Y3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ( ( member_real @ Y3 @ ( set_real2 @ Xs ) )
           => ( X3 = Y3 ) ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
           => ( X3 = Y ) )
       => ( ( foldr_real_real @ plus_plus_real @ Xs @ zero_zero_real )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_real @ Xs ) ) @ Y ) ) ) ) ).

% foldr_same
thf(fact_5739_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_5740_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_5741_foldr0,axiom,
    ! [Xs: list_real,C: real,D2: real] :
      ( ( foldr_real_real @ plus_plus_real @ Xs @ ( plus_plus_real @ C @ D2 ) )
      = ( plus_plus_real @ ( foldr_real_real @ plus_plus_real @ Xs @ D2 ) @ C ) ) ).

% foldr0
thf(fact_5742_foldr__one,axiom,
    ! [D2: nat,Ys: list_nat] : ( ord_less_eq_nat @ D2 @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D2 ) ) ).

% foldr_one
thf(fact_5743_foldr__same__int,axiom,
    ! [Xs: list_nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
           => ( X3 = Y3 ) ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( X3 = Y ) )
       => ( ( foldr_nat_nat @ plus_plus_nat @ Xs @ zero_zero_nat )
          = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ Y ) ) ) ) ).

% foldr_same_int
thf(fact_5744_foldr__mono,axiom,
    ! [Xs: list_nat,Ys: list_nat,C: nat,D2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( ord_less_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) )
       => ( ( ord_less_eq_nat @ C @ D2 )
         => ( ord_less_eq_nat @ ( plus_plus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs @ C ) @ ( size_size_list_nat @ Ys ) ) @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D2 ) ) ) ) ) ).

% foldr_mono
thf(fact_5745_round__0,axiom,
    ( ( archim8280529875227126926d_real @ zero_zero_real )
    = zero_zero_int ) ).

% round_0
thf(fact_5746_round__0,axiom,
    ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% round_0
thf(fact_5747_round__numeral,axiom,
    ! [N: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_5748_round__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_5749_foldr__length,axiom,
    ! [L: list_real] :
      ( ( foldr_real_nat
        @ ^ [X2: real] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_real @ L ) ) ).

% foldr_length
thf(fact_5750_foldr__length,axiom,
    ! [L: list_o] :
      ( ( foldr_o_nat
        @ ^ [X2: $o] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_o @ L ) ) ).

% foldr_length
thf(fact_5751_foldr__length,axiom,
    ! [L: list_nat] :
      ( ( foldr_nat_nat
        @ ^ [X2: nat] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_nat @ L ) ) ).

% foldr_length
thf(fact_5752_foldr__length,axiom,
    ! [L: list_int] :
      ( ( foldr_int_nat
        @ ^ [X2: int] : suc
        @ L
        @ zero_zero_nat )
      = ( size_size_list_int @ L ) ) ).

% foldr_length
thf(fact_5753_vebt__minti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_mint @ T ) ) ) ) ) ).

% vebt_minti_h
thf(fact_5754_vebt__maxti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_maxt @ T ) ) ) ) ) ).

% vebt_maxti_h
thf(fact_5755_vebt__mintilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ts2 ) )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) @ ( vEBT_vebt_minti @ ( nth_VEBT_VEBTi @ Tsi @ I ) )
        @ ^ [R5: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ Ts2 @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) ) ) ) ).

% vebt_mintilist
thf(fact_5756_vebt__maxtilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ts2 ) )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) @ ( vEBT_vebt_maxti @ ( nth_VEBT_VEBTi @ Tsi @ I ) )
        @ ^ [R5: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Ts2 @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) ) ) ) ).

% vebt_maxtilist
thf(fact_5757_foldr__cong,axiom,
    ! [A: nat,B: nat,L: list_o,K: list_o,F: $o > nat > nat,G: $o > nat > nat] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A6: nat,X3: $o] :
              ( ( member_o @ X3 @ ( set_o2 @ L ) )
             => ( ( F @ X3 @ A6 )
                = ( G @ X3 @ A6 ) ) )
         => ( ( foldr_o_nat @ F @ L @ A )
            = ( foldr_o_nat @ G @ K @ B ) ) ) ) ) ).

% foldr_cong
thf(fact_5758_foldr__cong,axiom,
    ! [A: real,B: real,L: list_real,K: list_real,F: real > real > real,G: real > real > real] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A6: real,X3: real] :
              ( ( member_real @ X3 @ ( set_real2 @ L ) )
             => ( ( F @ X3 @ A6 )
                = ( G @ X3 @ A6 ) ) )
         => ( ( foldr_real_real @ F @ L @ A )
            = ( foldr_real_real @ G @ K @ B ) ) ) ) ) ).

% foldr_cong
thf(fact_5759_foldr__cong,axiom,
    ! [A: nat,B: nat,L: list_nat,K: list_nat,F: nat > nat > nat,G: nat > nat > nat] :
      ( ( A = B )
     => ( ( L = K )
       => ( ! [A6: nat,X3: nat] :
              ( ( member_nat @ X3 @ ( set_nat2 @ L ) )
             => ( ( F @ X3 @ A6 )
                = ( G @ X3 @ A6 ) ) )
         => ( ( foldr_nat_nat @ F @ L @ A )
            = ( foldr_nat_nat @ G @ K @ B ) ) ) ) ) ).

% foldr_cong
thf(fact_5760_foldr__length__aux,axiom,
    ! [L: list_real,A: nat] :
      ( ( foldr_real_nat
        @ ^ [X2: real] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_real @ L ) ) ) ).

% foldr_length_aux
thf(fact_5761_foldr__length__aux,axiom,
    ! [L: list_o,A: nat] :
      ( ( foldr_o_nat
        @ ^ [X2: $o] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_o @ L ) ) ) ).

% foldr_length_aux
thf(fact_5762_foldr__length__aux,axiom,
    ! [L: list_nat,A: nat] :
      ( ( foldr_nat_nat
        @ ^ [X2: nat] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_nat @ L ) ) ) ).

% foldr_length_aux
thf(fact_5763_foldr__length__aux,axiom,
    ! [L: list_int,A: nat] :
      ( ( foldr_int_nat
        @ ^ [X2: int] : suc
        @ L
        @ A )
      = ( plus_plus_nat @ A @ ( size_size_list_int @ L ) ) ) ).

% foldr_length_aux
thf(fact_5764_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: $o > assn,Q5: $o > assn,C: heap_Time_Heap_o] :
      ( ( entails @ P @ P2 )
     => ( ! [X3: $o] : ( entails @ ( Q @ X3 ) @ ( Q5 @ X3 ) )
       => ( ( hoare_hoare_triple_o @ P2 @ C @ Q )
         => ( hoare_hoare_triple_o @ P @ C @ Q5 ) ) ) ) ).

% cons_rule
thf(fact_5765_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: option_nat > assn,Q5: option_nat > assn,C: heap_T2636463487746394924on_nat] :
      ( ( entails @ P @ P2 )
     => ( ! [X3: option_nat] : ( entails @ ( Q @ X3 ) @ ( Q5 @ X3 ) )
       => ( ( hoare_7629718768684598413on_nat @ P2 @ C @ Q )
         => ( hoare_7629718768684598413on_nat @ P @ C @ Q5 ) ) ) ) ).

% cons_rule
thf(fact_5766_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: nat > assn,Q5: nat > assn,C: heap_Time_Heap_nat] :
      ( ( entails @ P @ P2 )
     => ( ! [X3: nat] : ( entails @ ( Q @ X3 ) @ ( Q5 @ X3 ) )
       => ( ( hoare_3067605981109127869le_nat @ P2 @ C @ Q )
         => ( hoare_3067605981109127869le_nat @ P @ C @ Q5 ) ) ) ) ).

% cons_rule
thf(fact_5767_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: vEBT_VEBTi > assn,Q5: vEBT_VEBTi > assn,C: heap_T8145700208782473153_VEBTi] :
      ( ( entails @ P @ P2 )
     => ( ! [X3: vEBT_VEBTi] : ( entails @ ( Q @ X3 ) @ ( Q5 @ X3 ) )
       => ( ( hoare_1429296392585015714_VEBTi @ P2 @ C @ Q )
         => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q5 ) ) ) ) ).

% cons_rule
thf(fact_5768_cons__post__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,Q5: $o > assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( ! [X3: $o] : ( entails @ ( Q @ X3 ) @ ( Q5 @ X3 ) )
       => ( hoare_hoare_triple_o @ P @ C @ Q5 ) ) ) ).

% cons_post_rule
thf(fact_5769_cons__post__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,Q5: option_nat > assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( ! [X3: option_nat] : ( entails @ ( Q @ X3 ) @ ( Q5 @ X3 ) )
       => ( hoare_7629718768684598413on_nat @ P @ C @ Q5 ) ) ) ).

% cons_post_rule
thf(fact_5770_cons__post__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,Q5: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( ! [X3: nat] : ( entails @ ( Q @ X3 ) @ ( Q5 @ X3 ) )
       => ( hoare_3067605981109127869le_nat @ P @ C @ Q5 ) ) ) ).

% cons_post_rule
thf(fact_5771_cons__post__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,Q5: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( ! [X3: vEBT_VEBTi] : ( entails @ ( Q @ X3 ) @ ( Q5 @ X3 ) )
       => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q5 ) ) ) ).

% cons_post_rule
thf(fact_5772_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,R3: assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ P @ R3 ) @ C
        @ ^ [X2: $o] : ( times_times_assn @ ( Q @ X2 ) @ R3 ) ) ) ).

% frame_rule
thf(fact_5773_frame__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,R3: assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( hoare_7629718768684598413on_nat @ ( times_times_assn @ P @ R3 ) @ C
        @ ^ [X2: option_nat] : ( times_times_assn @ ( Q @ X2 ) @ R3 ) ) ) ).

% frame_rule
thf(fact_5774_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,R3: assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( hoare_3067605981109127869le_nat @ ( times_times_assn @ P @ R3 ) @ C
        @ ^ [X2: nat] : ( times_times_assn @ ( Q @ X2 ) @ R3 ) ) ) ).

% frame_rule
thf(fact_5775_frame__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,R3: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ R3 ) @ C
        @ ^ [X2: vEBT_VEBTi] : ( times_times_assn @ ( Q @ X2 ) @ R3 ) ) ) ).

% frame_rule
thf(fact_5776_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ P2 )
     => ( ( hoare_hoare_triple_o @ P2 @ C @ Q )
       => ( hoare_hoare_triple_o @ P @ C @ Q ) ) ) ).

% cons_pre_rule
thf(fact_5777_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( entails @ P @ P2 )
     => ( ( hoare_7629718768684598413on_nat @ P2 @ C @ Q )
       => ( hoare_7629718768684598413on_nat @ P @ C @ Q ) ) ) ).

% cons_pre_rule
thf(fact_5778_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( entails @ P @ P2 )
     => ( ( hoare_3067605981109127869le_nat @ P2 @ C @ Q )
       => ( hoare_3067605981109127869le_nat @ P @ C @ Q ) ) ) ).

% cons_pre_rule
thf(fact_5779_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( entails @ P @ P2 )
     => ( ( hoare_1429296392585015714_VEBTi @ P2 @ C @ Q )
       => ( hoare_1429296392585015714_VEBTi @ P @ C @ Q ) ) ) ).

% cons_pre_rule
thf(fact_5780_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_5781_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_5782_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_5783_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_5784_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_5785_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_5786_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( B
       => ( hoare_hoare_triple_o @ P @ F @ Q ) )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_5787_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( B
       => ( hoare_7629718768684598413on_nat @ P @ F @ Q ) )
     => ( hoare_7629718768684598413on_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_5788_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( B
       => ( hoare_3067605981109127869le_nat @ P @ F @ Q ) )
     => ( hoare_3067605981109127869le_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_5789_norm__pre__pure__rule1,axiom,
    ! [B: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( B
       => ( hoare_1429296392585015714_VEBTi @ P @ F @ Q ) )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_5790_lowi__h,axiom,
    ! [X: nat,N: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X @ N )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_low @ X @ N ) ) ) ) ).

% lowi_h
thf(fact_5791_highi__h,axiom,
    ! [X: nat,N: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X @ N )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_high @ X @ N ) ) ) ) ).

% highi_h
thf(fact_5792_builupicorr,axiom,
    ! [N: nat] : ( hoare_1429296392585015714_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) ) ).

% builupicorr
thf(fact_5793_builupi_Hcorr,axiom,
    ! [N: nat] : ( hoare_1429296392585015714_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) ) ).

% builupi'corr
thf(fact_5794_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_VEBT_VEBT,F: vEBT_VEBT > real,Bound: real,I: real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_s6755466524823107622T_VEBT @ Xs ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5795_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_real,F: real > real,Bound: real,I: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_real @ Xs ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5796_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_o,F: $o > real,Bound: real,I: real] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_o_real @ F @ Xs ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_o @ Xs ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5797_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_nat,F: nat > real,Bound: real,I: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_nat @ Xs ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5798_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_int,F: int > real,Bound: real,I: real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ F @ Xs ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_int @ Xs ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5799_f__g__map__foldr__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: vEBT_VEBT > real,C: real,G: vEBT_VEBT > real,D2: real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs ) @ D2 ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ G @ Xs ) @ zero_zero_real ) ) @ D2 ) ) ) ).

% f_g_map_foldr_bound
thf(fact_5800_f__g__map__foldr__bound,axiom,
    ! [Xs: list_real,F: real > real,C: real,G: real > real,D2: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs ) @ D2 ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ G @ Xs ) @ zero_zero_real ) ) @ D2 ) ) ) ).

% f_g_map_foldr_bound
thf(fact_5801_f__g__map__foldr__bound,axiom,
    ! [Xs: list_nat,F: nat > real,C: real,G: nat > real,D2: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs ) @ D2 ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ G @ Xs ) @ zero_zero_real ) ) @ D2 ) ) ) ).

% f_g_map_foldr_bound
thf(fact_5802_f__g__map__foldr__bound,axiom,
    ! [Xs: list_int,F: int > real,C: real,G: int > real,D2: real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ F @ Xs ) @ D2 ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ G @ Xs ) @ zero_zero_real ) ) @ D2 ) ) ) ).

% f_g_map_foldr_bound
thf(fact_5803_map__ident,axiom,
    ( ( map_nat_nat
      @ ^ [X2: nat] : X2 )
    = ( ^ [Xs2: list_nat] : Xs2 ) ) ).

% map_ident
thf(fact_5804_listsum__bound,axiom,
    ! [Xs: list_complex,F: complex > real,Y: real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_complex_real @ F @ Xs ) @ Y ) ) ) ).

% listsum_bound
thf(fact_5805_listsum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: vEBT_VEBT > real,Y: real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs ) @ Y ) ) ) ).

% listsum_bound
thf(fact_5806_listsum__bound,axiom,
    ! [Xs: list_real,F: real > real,Y: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs ) @ Y ) ) ) ).

% listsum_bound
thf(fact_5807_listsum__bound,axiom,
    ! [Xs: list_nat,F: nat > real,Y: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs ) @ Y ) ) ) ).

% listsum_bound
thf(fact_5808_listsum__bound,axiom,
    ! [Xs: list_int,F: int > real,Y: real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ F @ Xs ) @ Y ) ) ) ).

% listsum_bound
thf(fact_5809_length__map,axiom,
    ! [F: vEBT_VEBT > real,Xs: list_VEBT_VEBT] :
      ( ( size_size_list_real @ ( map_VEBT_VEBT_real @ F @ Xs ) )
      = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% length_map
thf(fact_5810_length__map,axiom,
    ! [F: real > real,Xs: list_real] :
      ( ( size_size_list_real @ ( map_real_real @ F @ Xs ) )
      = ( size_size_list_real @ Xs ) ) ).

% length_map
thf(fact_5811_length__map,axiom,
    ! [F: $o > real,Xs: list_o] :
      ( ( size_size_list_real @ ( map_o_real @ F @ Xs ) )
      = ( size_size_list_o @ Xs ) ) ).

% length_map
thf(fact_5812_length__map,axiom,
    ! [F: nat > real,Xs: list_nat] :
      ( ( size_size_list_real @ ( map_nat_real @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_5813_length__map,axiom,
    ! [F: int > real,Xs: list_int] :
      ( ( size_size_list_real @ ( map_int_real @ F @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_map
thf(fact_5814_length__map,axiom,
    ! [F: real > $o,Xs: list_real] :
      ( ( size_size_list_o @ ( map_real_o @ F @ Xs ) )
      = ( size_size_list_real @ Xs ) ) ).

% length_map
thf(fact_5815_length__map,axiom,
    ! [F: $o > $o,Xs: list_o] :
      ( ( size_size_list_o @ ( map_o_o @ F @ Xs ) )
      = ( size_size_list_o @ Xs ) ) ).

% length_map
thf(fact_5816_length__map,axiom,
    ! [F: nat > $o,Xs: list_nat] :
      ( ( size_size_list_o @ ( map_nat_o @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_5817_length__map,axiom,
    ! [F: int > $o,Xs: list_int] :
      ( ( size_size_list_o @ ( map_int_o @ F @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_map
thf(fact_5818_length__map,axiom,
    ! [F: vEBT_VEBT > nat,Xs: list_VEBT_VEBT] :
      ( ( size_size_list_nat @ ( map_VEBT_VEBT_nat @ F @ Xs ) )
      = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% length_map
thf(fact_5819_map__eq__conv,axiom,
    ! [F: vEBT_VEBT > real,Xs: list_VEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs )
        = ( map_VEBT_VEBT_real @ G @ Xs ) )
      = ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_5820_map__eq__conv,axiom,
    ! [F: vEBT_VEBT > nat,Xs: list_VEBT_VEBT,G: vEBT_VEBT > nat] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs )
        = ( map_VEBT_VEBT_nat @ G @ Xs ) )
      = ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_5821_map__eq__conv,axiom,
    ! [F: nat > $o,Xs: list_nat,G: nat > $o] :
      ( ( ( map_nat_o @ F @ Xs )
        = ( map_nat_o @ G @ Xs ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_5822_map__eq__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Xs ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) ) ) ) ).

% map_eq_conv
thf(fact_5823_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_VEBT_VEBT @ ( map_VE8901447254227204932T_VEBT @ F @ Xs ) @ N )
        = ( F @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_5824_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,F: vEBT_VEBTi > vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( nth_VEBT_VEBT @ ( map_VE7998069337340375161T_VEBT @ F @ Xs ) @ N )
        = ( F @ ( nth_VEBT_VEBTi @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_5825_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBTi] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_VEBT_VEBTi @ ( map_VE7029150624388687525_VEBTi @ F @ Xs ) @ N )
        = ( F @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_5826_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,F: vEBT_VEBTi > vEBT_VEBTi] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( nth_VEBT_VEBTi @ ( map_VE483055756984248624_VEBTi @ F @ Xs ) @ N )
        = ( F @ ( nth_VEBT_VEBTi @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_5827_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,F: vEBT_VEBTi > nat] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( nth_nat @ ( map_VEBT_VEBTi_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_VEBT_VEBTi @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_5828_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: vEBT_VEBT > int] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_int @ ( map_VEBT_VEBT_int @ F @ Xs ) @ N )
        = ( F @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_5829_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,F: vEBT_VEBTi > int] :
      ( ( ord_less_nat @ N @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( ( nth_int @ ( map_VEBT_VEBTi_int @ F @ Xs ) @ N )
        = ( F @ ( nth_VEBT_VEBTi @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_5830_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_real @ ( map_VEBT_VEBT_real @ F @ Xs ) @ N )
        = ( F @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_5831_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_nat @ ( map_VEBT_VEBT_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_5832_nth__map,axiom,
    ! [N: nat,Xs: list_real,F: real > vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
     => ( ( nth_VEBT_VEBT @ ( map_real_VEBT_VEBT @ F @ Xs ) @ N )
        = ( F @ ( nth_real @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_5833_list_Omap__ident,axiom,
    ! [T: list_nat] :
      ( ( map_nat_nat
        @ ^ [X2: nat] : X2
        @ T )
      = T ) ).

% list.map_ident
thf(fact_5834_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs: list_VEBT_VEBT,G: vEBT_VEBT > real,Ys: list_VEBT_VEBT] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs )
        = ( map_VEBT_VEBT_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5835_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs: list_VEBT_VEBT,G: vEBT_VEBT > nat,Ys: list_VEBT_VEBT] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs )
        = ( map_VEBT_VEBT_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5836_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs: list_VEBT_VEBT,G: real > real,Ys: list_real] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs )
        = ( map_real_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_real @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5837_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs: list_VEBT_VEBT,G: real > nat,Ys: list_real] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs )
        = ( map_real_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_real @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5838_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs: list_VEBT_VEBT,G: $o > real,Ys: list_o] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs )
        = ( map_o_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5839_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs: list_VEBT_VEBT,G: $o > nat,Ys: list_o] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs )
        = ( map_o_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5840_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs: list_VEBT_VEBT,G: nat > real,Ys: list_nat] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs )
        = ( map_nat_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5841_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs: list_VEBT_VEBT,G: nat > nat,Ys: list_nat] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5842_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs: list_VEBT_VEBT,G: int > real,Ys: list_int] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs )
        = ( map_int_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_int @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5843_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs: list_VEBT_VEBT,G: int > nat,Ys: list_int] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs )
        = ( map_int_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_int @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5844_list_Omap__cong,axiom,
    ! [X: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( X = Ya )
     => ( ! [Z6: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Z6 @ ( set_VEBT_VEBT2 @ Ya ) )
           => ( ( F @ Z6 )
              = ( G @ Z6 ) ) )
       => ( ( map_VEBT_VEBT_real @ F @ X )
          = ( map_VEBT_VEBT_real @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_5845_list_Omap__cong,axiom,
    ! [X: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( X = Ya )
     => ( ! [Z6: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Z6 @ ( set_VEBT_VEBT2 @ Ya ) )
           => ( ( F @ Z6 )
              = ( G @ Z6 ) ) )
       => ( ( map_VEBT_VEBT_nat @ F @ X )
          = ( map_VEBT_VEBT_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_5846_list_Omap__cong,axiom,
    ! [X: list_nat,Ya: list_nat,F: nat > $o,G: nat > $o] :
      ( ( X = Ya )
     => ( ! [Z6: nat] :
            ( ( member_nat @ Z6 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z6 )
              = ( G @ Z6 ) ) )
       => ( ( map_nat_o @ F @ X )
          = ( map_nat_o @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_5847_list_Omap__cong,axiom,
    ! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
      ( ( X = Ya )
     => ( ! [Z6: nat] :
            ( ( member_nat @ Z6 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z6 )
              = ( G @ Z6 ) ) )
       => ( ( map_nat_nat @ F @ X )
          = ( map_nat_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_5848_list_Omap__cong0,axiom,
    ! [X: list_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ! [Z6: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z6 @ ( set_VEBT_VEBT2 @ X ) )
         => ( ( F @ Z6 )
            = ( G @ Z6 ) ) )
     => ( ( map_VEBT_VEBT_real @ F @ X )
        = ( map_VEBT_VEBT_real @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_5849_list_Omap__cong0,axiom,
    ! [X: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [Z6: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z6 @ ( set_VEBT_VEBT2 @ X ) )
         => ( ( F @ Z6 )
            = ( G @ Z6 ) ) )
     => ( ( map_VEBT_VEBT_nat @ F @ X )
        = ( map_VEBT_VEBT_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_5850_list_Omap__cong0,axiom,
    ! [X: list_nat,F: nat > $o,G: nat > $o] :
      ( ! [Z6: nat] :
          ( ( member_nat @ Z6 @ ( set_nat2 @ X ) )
         => ( ( F @ Z6 )
            = ( G @ Z6 ) ) )
     => ( ( map_nat_o @ F @ X )
        = ( map_nat_o @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_5851_list_Omap__cong0,axiom,
    ! [X: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [Z6: nat] :
          ( ( member_nat @ Z6 @ ( set_nat2 @ X ) )
         => ( ( F @ Z6 )
            = ( G @ Z6 ) ) )
     => ( ( map_nat_nat @ F @ X )
        = ( map_nat_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_5852_list_Oinj__map__strong,axiom,
    ! [X: list_VEBT_VEBT,Xa3: list_VEBT_VEBT,F: vEBT_VEBT > real,Fa: vEBT_VEBT > real] :
      ( ! [Z6: vEBT_VEBT,Za: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z6 @ ( set_VEBT_VEBT2 @ X ) )
         => ( ( member_VEBT_VEBT @ Za @ ( set_VEBT_VEBT2 @ Xa3 ) )
           => ( ( ( F @ Z6 )
                = ( Fa @ Za ) )
             => ( Z6 = Za ) ) ) )
     => ( ( ( map_VEBT_VEBT_real @ F @ X )
          = ( map_VEBT_VEBT_real @ Fa @ Xa3 ) )
       => ( X = Xa3 ) ) ) ).

% list.inj_map_strong
thf(fact_5853_list_Oinj__map__strong,axiom,
    ! [X: list_VEBT_VEBT,Xa3: list_VEBT_VEBT,F: vEBT_VEBT > nat,Fa: vEBT_VEBT > nat] :
      ( ! [Z6: vEBT_VEBT,Za: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z6 @ ( set_VEBT_VEBT2 @ X ) )
         => ( ( member_VEBT_VEBT @ Za @ ( set_VEBT_VEBT2 @ Xa3 ) )
           => ( ( ( F @ Z6 )
                = ( Fa @ Za ) )
             => ( Z6 = Za ) ) ) )
     => ( ( ( map_VEBT_VEBT_nat @ F @ X )
          = ( map_VEBT_VEBT_nat @ Fa @ Xa3 ) )
       => ( X = Xa3 ) ) ) ).

% list.inj_map_strong
thf(fact_5854_list_Oinj__map__strong,axiom,
    ! [X: list_nat,Xa3: list_nat,F: nat > $o,Fa: nat > $o] :
      ( ! [Z6: nat,Za: nat] :
          ( ( member_nat @ Z6 @ ( set_nat2 @ X ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa3 ) )
           => ( ( ( F @ Z6 )
                = ( Fa @ Za ) )
             => ( Z6 = Za ) ) ) )
     => ( ( ( map_nat_o @ F @ X )
          = ( map_nat_o @ Fa @ Xa3 ) )
       => ( X = Xa3 ) ) ) ).

% list.inj_map_strong
thf(fact_5855_list_Oinj__map__strong,axiom,
    ! [X: list_nat,Xa3: list_nat,F: nat > nat,Fa: nat > nat] :
      ( ! [Z6: nat,Za: nat] :
          ( ( member_nat @ Z6 @ ( set_nat2 @ X ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa3 ) )
           => ( ( ( F @ Z6 )
                = ( Fa @ Za ) )
             => ( Z6 = Za ) ) ) )
     => ( ( ( map_nat_nat @ F @ X )
          = ( map_nat_nat @ Fa @ Xa3 ) )
       => ( X = Xa3 ) ) ) ).

% list.inj_map_strong
thf(fact_5856_map__ext,axiom,
    ! [Xs: list_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_VEBT_VEBT_real @ F @ Xs )
        = ( map_VEBT_VEBT_real @ G @ Xs ) ) ) ).

% map_ext
thf(fact_5857_map__ext,axiom,
    ! [Xs: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_VEBT_VEBT_nat @ F @ Xs )
        = ( map_VEBT_VEBT_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_5858_map__ext,axiom,
    ! [Xs: list_nat,F: nat > $o,G: nat > $o] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_nat_o @ F @ Xs )
        = ( map_nat_o @ G @ Xs ) ) ) ).

% map_ext
thf(fact_5859_map__ext,axiom,
    ! [Xs: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_5860_map__idI,axiom,
    ! [Xs: list_complex,F: complex > complex] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_complex_complex @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_5861_map__idI,axiom,
    ! [Xs: list_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_VE8901447254227204932T_VEBT @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_5862_map__idI,axiom,
    ! [Xs: list_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_real_real @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_5863_map__idI,axiom,
    ! [Xs: list_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_nat_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_5864_map__idI,axiom,
    ! [Xs: list_int,F: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_int_int @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_5865_map__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( Xs = Ys )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_VEBT_VEBT_real @ F @ Xs )
          = ( map_VEBT_VEBT_real @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_5866_map__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( Xs = Ys )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_VEBT_VEBT_nat @ F @ Xs )
          = ( map_VEBT_VEBT_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_5867_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > $o,G: nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_nat_o @ F @ Xs )
          = ( map_nat_o @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_5868_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
      ( ( Xs = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_nat_nat @ F @ Xs )
          = ( map_nat_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_5869_ex__map__conv,axiom,
    ! [Ys: list_o,F: nat > $o] :
      ( ( ? [Xs2: list_nat] :
            ( Ys
            = ( map_nat_o @ F @ Xs2 ) ) )
      = ( ! [X2: $o] :
            ( ( member_o @ X2 @ ( set_o2 @ Ys ) )
           => ? [Y5: nat] :
                ( X2
                = ( F @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_5870_ex__map__conv,axiom,
    ! [Ys: list_real,F: vEBT_VEBT > real] :
      ( ( ? [Xs2: list_VEBT_VEBT] :
            ( Ys
            = ( map_VEBT_VEBT_real @ F @ Xs2 ) ) )
      = ( ! [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ Ys ) )
           => ? [Y5: vEBT_VEBT] :
                ( X2
                = ( F @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_5871_ex__map__conv,axiom,
    ! [Ys: list_nat,F: vEBT_VEBT > nat] :
      ( ( ? [Xs2: list_VEBT_VEBT] :
            ( Ys
            = ( map_VEBT_VEBT_nat @ F @ Xs2 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
           => ? [Y5: vEBT_VEBT] :
                ( X2
                = ( F @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_5872_ex__map__conv,axiom,
    ! [Ys: list_nat,F: nat > nat] :
      ( ( ? [Xs2: list_nat] :
            ( Ys
            = ( map_nat_nat @ F @ Xs2 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
           => ? [Y5: nat] :
                ( X2
                = ( F @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_5873_map__update,axiom,
    ! [F: nat > $o,Xs: list_nat,K: nat,Y: nat] :
      ( ( map_nat_o @ F @ ( list_update_nat @ Xs @ K @ Y ) )
      = ( list_update_o @ ( map_nat_o @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_5874_map__update,axiom,
    ! [F: nat > nat,Xs: list_nat,K: nat,Y: nat] :
      ( ( map_nat_nat @ F @ ( list_update_nat @ Xs @ K @ Y ) )
      = ( list_update_nat @ ( map_nat_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_5875_map__update,axiom,
    ! [F: vEBT_VEBT > real,Xs: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] :
      ( ( map_VEBT_VEBT_real @ F @ ( list_u1324408373059187874T_VEBT @ Xs @ K @ Y ) )
      = ( list_update_real @ ( map_VEBT_VEBT_real @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_5876_map__update,axiom,
    ! [F: vEBT_VEBT > nat,Xs: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat @ F @ ( list_u1324408373059187874T_VEBT @ Xs @ K @ Y ) )
      = ( list_update_nat @ ( map_VEBT_VEBT_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_5877_map__update,axiom,
    ! [F: vEBT_VEBT > vEBT_VEBT,Xs: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] :
      ( ( map_VE8901447254227204932T_VEBT @ F @ ( list_u1324408373059187874T_VEBT @ Xs @ K @ Y ) )
      = ( list_u1324408373059187874T_VEBT @ ( map_VE8901447254227204932T_VEBT @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_5878_map__update,axiom,
    ! [F: vEBT_VEBT > vEBT_VEBTi,Xs: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] :
      ( ( map_VE7029150624388687525_VEBTi @ F @ ( list_u1324408373059187874T_VEBT @ Xs @ K @ Y ) )
      = ( list_u6098035379799741383_VEBTi @ ( map_VE7029150624388687525_VEBTi @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_5879_map__update,axiom,
    ! [F: vEBT_VEBTi > vEBT_VEBT,Xs: list_VEBT_VEBTi,K: nat,Y: vEBT_VEBTi] :
      ( ( map_VE7998069337340375161T_VEBT @ F @ ( list_u6098035379799741383_VEBTi @ Xs @ K @ Y ) )
      = ( list_u1324408373059187874T_VEBT @ ( map_VE7998069337340375161T_VEBT @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_5880_map__update,axiom,
    ! [F: vEBT_VEBTi > vEBT_VEBTi,Xs: list_VEBT_VEBTi,K: nat,Y: vEBT_VEBTi] :
      ( ( map_VE483055756984248624_VEBTi @ F @ ( list_u6098035379799741383_VEBTi @ Xs @ K @ Y ) )
      = ( list_u6098035379799741383_VEBTi @ ( map_VE483055756984248624_VEBTi @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_5881_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: vEBT_VEBT > real,X: vEBT_VEBT] :
      ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
       => ( ( F @ ( nth_VEBT_VEBT @ L @ I ) )
          = ( F @ X ) ) )
     => ( ( map_VEBT_VEBT_real @ F @ ( list_u1324408373059187874T_VEBT @ L @ I @ X ) )
        = ( map_VEBT_VEBT_real @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_5882_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F: vEBT_VEBT > nat,X: vEBT_VEBT] :
      ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L ) )
       => ( ( F @ ( nth_VEBT_VEBT @ L @ I ) )
          = ( F @ X ) ) )
     => ( ( map_VEBT_VEBT_nat @ F @ ( list_u1324408373059187874T_VEBT @ L @ I @ X ) )
        = ( map_VEBT_VEBT_nat @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_5883_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: nat > $o,X: nat] :
      ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
       => ( ( F @ ( nth_nat @ L @ I ) )
          = ( F @ X ) ) )
     => ( ( map_nat_o @ F @ ( list_update_nat @ L @ I @ X ) )
        = ( map_nat_o @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_5884_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,F: nat > nat,X: nat] :
      ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
       => ( ( F @ ( nth_nat @ L @ I ) )
          = ( F @ X ) ) )
     => ( ( map_nat_nat @ F @ ( list_update_nat @ L @ I @ X ) )
        = ( map_nat_nat @ F @ L ) ) ) ).

% map_upd_eq
thf(fact_5885_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_cnt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList ) @ zero_zero_real ) ) ) ).

% VEBT_internal.cnt.simps(2)
thf(fact_5886_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( ( vEBT_VEBT_cnt @ X )
        = Y )
     => ( ( ? [A6: $o,B5: $o] :
              ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
         => ( Y != one_one_real ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList2 ) @ zero_zero_real ) ) ) ) ) ) ).

% VEBT_internal.cnt.elims
thf(fact_5887_vebt__buildupi__rule,axiom,
    ! [N: nat] : ( time_htt_VEBT_VEBTi @ ( pure_assn @ ( ord_less_nat @ zero_zero_nat @ N ) ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% vebt_buildupi_rule
thf(fact_5888_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% htt_vebt_buildupi'_univ
thf(fact_5889_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% htt_vebt_buildupi_univ
thf(fact_5890_vebt__maxti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_maxt @ T ) ) ) )
      @ one_one_nat ) ).

% vebt_maxti_hT
thf(fact_5891_vebt__minti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_mint @ T ) ) ) )
      @ one_one_nat ) ).

% vebt_minti_hT
thf(fact_5892_lowi__hT,axiom,
    ! [X: nat,N: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X @ N )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_low @ X @ N ) ) )
      @ one_one_nat ) ).

% lowi_hT
thf(fact_5893_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F: vEBT_VEBT > nat,Bound: nat,I: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ F @ Xs ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_5894_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_real,F: real > nat,Bound: nat,I: nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_real_nat @ F @ Xs ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_real @ Xs ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_5895_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_o,F: $o > nat,Bound: nat,I: nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_o_nat @ F @ Xs ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_5896_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_nat,F: nat > nat,Bound: nat,I: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_nat_nat @ F @ Xs ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_5897_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_int,F: int > nat,Bound: nat,I: nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_int_nat @ F @ Xs ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_5898_real__nat__list,axiom,
    ! [F: vEBT_VEBT > nat,Xs: list_VEBT_VEBT,C: nat] :
      ( ( semiri5074537144036343181t_real @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ F @ Xs ) @ C ) )
      = ( foldr_real_real @ plus_plus_real
        @ ( map_VEBT_VEBT_real
          @ ^ [X2: vEBT_VEBT] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
          @ Xs )
        @ ( semiri5074537144036343181t_real @ C ) ) ) ).

% real_nat_list
thf(fact_5899_real__nat__list,axiom,
    ! [F: nat > nat,Xs: list_nat,C: nat] :
      ( ( semiri5074537144036343181t_real @ ( foldr_nat_nat @ plus_plus_nat @ ( map_nat_nat @ F @ Xs ) @ C ) )
      = ( foldr_real_real @ plus_plus_real
        @ ( map_nat_real
          @ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
          @ Xs )
        @ ( semiri5074537144036343181t_real @ C ) ) ) ).

% real_nat_list
thf(fact_5900_htt__vebt__buildupi_H,axiom,
    ! [N: nat] : ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).

% htt_vebt_buildupi'
thf(fact_5901_htt__vebt__buildupi,axiom,
    ! [N: nat] : ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N ) ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).

% htt_vebt_buildupi
thf(fact_5902_highi__hT,axiom,
    ! [X: nat,N: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X @ N )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_high @ X @ N ) ) )
      @ one_one_nat ) ).

% highi_hT
thf(fact_5903_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_cnt2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.cnt'.simps(2)
thf(fact_5904_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X )
        = Y )
     => ( ( ? [A6: $o,B5: $o] :
              ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
         => ( Y != one_one_nat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList2 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.cnt'.elims
thf(fact_5905_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_space2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.space'.simps(2)
thf(fact_5906_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space2 @ X )
        = Y )
     => ( ( ? [A6: $o,B5: $o] :
              ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
         => ( Y
           != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList2 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.space'.elims
thf(fact_5907_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_space @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.space.simps(2)
thf(fact_5908_VEBT__internal_Ospace_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space @ X )
        = Y )
     => ( ( ? [A6: $o,B5: $o] :
              ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList2 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.space.elims
thf(fact_5909_T__vebt__buildupi,axiom,
    ! [N: nat,H2: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ N ) @ H2 ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).

% T_vebt_buildupi
thf(fact_5910_lowi__def,axiom,
    ( vEBT_VEBT_lowi
    = ( ^ [X2: nat,N4: nat] : ( heap_Time_return_nat @ ( modulo_modulo_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% lowi_def
thf(fact_5911_highi__def,axiom,
    ( vEBT_VEBT_highi
    = ( ^ [X2: nat,N4: nat] : ( heap_Time_return_nat @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% highi_def
thf(fact_5912_TBOUND__buildupi,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( time_T5737551269749752165_VEBTi @ ( vEBT_vebt_buildupi @ N ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% TBOUND_buildupi
thf(fact_5913_lowsimp,axiom,
    ! [X: nat,N: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_low @ X @ N ) )
      = ( vEBT_VEBT_lowi @ X @ N ) ) ).

% lowsimp
thf(fact_5914_highsimp,axiom,
    ! [X: nat,N: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_high @ X @ N ) )
      = ( vEBT_VEBT_highi @ X @ N ) ) ).

% highsimp
thf(fact_5915_TBOUND__vebt__buildupi,axiom,
    ! [N: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V739175172307565963ildupi @ N ) @ ( vEBT_V441764108873111860ildupi @ N ) ) ).

% TBOUND_vebt_buildupi
thf(fact_5916_return__sp__rule,axiom,
    ! [P: assn,X: $o] :
      ( hoare_hoare_triple_o @ P @ ( heap_Time_return_o @ X )
      @ ^ [R5: $o] : ( times_times_assn @ P @ ( pure_assn @ ( R5 = X ) ) ) ) ).

% return_sp_rule
thf(fact_5917_return__sp__rule,axiom,
    ! [P: assn,X: option_nat] :
      ( hoare_7629718768684598413on_nat @ P @ ( heap_T3487192422709364219on_nat @ X )
      @ ^ [R5: option_nat] : ( times_times_assn @ P @ ( pure_assn @ ( R5 = X ) ) ) ) ).

% return_sp_rule
thf(fact_5918_return__sp__rule,axiom,
    ! [P: assn,X: nat] :
      ( hoare_3067605981109127869le_nat @ P @ ( heap_Time_return_nat @ X )
      @ ^ [R5: nat] : ( times_times_assn @ P @ ( pure_assn @ ( R5 = X ) ) ) ) ).

% return_sp_rule
thf(fact_5919_return__sp__rule,axiom,
    ! [P: assn,X: vEBT_VEBTi] :
      ( hoare_1429296392585015714_VEBTi @ P @ ( heap_T3630416162098727440_VEBTi @ X )
      @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ P @ ( pure_assn @ ( R5 = X ) ) ) ) ).

% return_sp_rule
thf(fact_5920_time__replicate,axiom,
    ! [X: heap_T8145700208782473153_VEBTi,C: nat,N: nat,H2: heap_e7401611519738050253t_unit] :
      ( ! [H3: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ X @ H3 ) @ C )
     => ( ord_less_eq_nat @ ( time_t3534373299052942712_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N @ X ) @ H2 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N ) ) ) ) ).

% time_replicate
thf(fact_5921_TBOUND__replicate,axiom,
    ! [X: heap_T8145700208782473153_VEBTi,C: nat,N: nat] :
      ( ( time_T5737551269749752165_VEBTi @ X @ C )
     => ( time_T8149879359713347829_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N @ X ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N ) ) ) ) ).

% TBOUND_replicate
thf(fact_5922_TBOUND__replicate,axiom,
    ! [X: heap_Time_Heap_nat,C: nat,N: nat] :
      ( ( time_TBOUND_nat @ X @ C )
     => ( time_TBOUND_list_nat @ ( vEBT_V7726092123322077554ei_nat @ N @ X ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N ) ) ) ) ).

% TBOUND_replicate
thf(fact_5923_TBOUND__replicate,axiom,
    ! [X: heap_T2636463487746394924on_nat,C: nat,N: nat] :
      ( ( time_T8353473612707095248on_nat @ X @ C )
     => ( time_T3808005469503390304on_nat @ ( vEBT_V792416675989592002on_nat @ N @ X ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N ) ) ) ) ).

% TBOUND_replicate
thf(fact_5924_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_5925_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_5926_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_5927_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_5928_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_5929_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_5930_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_5931_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_5932_TBOUND__lowi,axiom,
    ! [X: nat,N: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_lowi @ X @ N ) @ one_one_nat ) ).

% TBOUND_lowi
thf(fact_5933_TBOUND__highi,axiom,
    ! [X: nat,N: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_highi @ X @ N ) @ one_one_nat ) ).

% TBOUND_highi
thf(fact_5934_TBOUND__vebt__minti,axiom,
    ! [T: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_minti @ T ) @ one_one_nat ) ).

% TBOUND_vebt_minti
thf(fact_5935_TBOUND__vebt__maxti,axiom,
    ! [T: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_maxti @ T ) @ one_one_nat ) ).

% TBOUND_vebt_maxti
thf(fact_5936_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ M @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_5937_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5055182867167087721od_nat @ M @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_5938_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_5939_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_5940_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_5941_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
      = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_5942_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_5943_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_5944_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_5945_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_5946_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_5947_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_5948_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N4: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) ) ) ) ).

% divmod_int_def
thf(fact_5949_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = ( heap_T3487192422709364219on_nat @ none_nat ) ) ).

% vebt_maxti.simps(2)
thf(fact_5950_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = ( heap_T3487192422709364219on_nat @ none_nat ) ) ).

% vebt_minti.simps(2)
thf(fact_5951_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N4: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N4 ) ) ) ) ) ).

% divmod_def
thf(fact_5952_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N4: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N4 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N4 ) ) ) ) ) ).

% divmod_def
thf(fact_5953_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N4: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N4 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N4 ) ) ) ) ) ).

% divmod_def
thf(fact_5954_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N4: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N4 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N4 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_5955_vebt__minti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi ) ) ) ).

% vebt_minti.simps(3)
thf(fact_5956_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma ) ) ) ).

% vebt_maxti.simps(3)
thf(fact_5957_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N4: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M5 @ N4 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M5 ) ) @ ( unique5026877609467782581ep_nat @ N4 @ ( unique5055182867167087721od_nat @ M5 @ ( bit0 @ N4 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5958_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N4: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M5 @ N4 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M5 ) ) @ ( unique5024387138958732305ep_int @ N4 @ ( unique5052692396658037445od_int @ M5 @ ( bit0 @ N4 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5959_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N4: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M5 @ N4 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M5 ) ) @ ( unique4921790084139445826nteger @ N4 @ ( unique3479559517661332726nteger @ M5 @ ( bit0 @ N4 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5960_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_5961_htt__cons__rule,axiom,
    ! [P2: assn,C: heap_T8145700208782473153_VEBTi,Q5: vEBT_VEBTi > assn,T3: nat,P: assn,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ P2 @ C @ Q5 @ T3 )
     => ( ( entails @ P @ P2 )
       => ( ! [X3: vEBT_VEBTi] : ( entails @ ( Q5 @ X3 ) @ ( Q @ X3 ) )
         => ( ( ord_less_eq_nat @ T3 @ T )
           => ( time_htt_VEBT_VEBTi @ P @ C @ Q @ T ) ) ) ) ) ).

% htt_cons_rule
thf(fact_5962_htt__cons__rule,axiom,
    ! [P2: assn,C: heap_T2636463487746394924on_nat,Q5: option_nat > assn,T3: nat,P: assn,Q: option_nat > assn,T: nat] :
      ( ( time_htt_option_nat @ P2 @ C @ Q5 @ T3 )
     => ( ( entails @ P @ P2 )
       => ( ! [X3: option_nat] : ( entails @ ( Q5 @ X3 ) @ ( Q @ X3 ) )
         => ( ( ord_less_eq_nat @ T3 @ T )
           => ( time_htt_option_nat @ P @ C @ Q @ T ) ) ) ) ) ).

% htt_cons_rule
thf(fact_5963_htt__cons__rule,axiom,
    ! [P2: assn,C: heap_Time_Heap_nat,Q5: nat > assn,T3: nat,P: assn,Q: nat > assn,T: nat] :
      ( ( time_htt_nat @ P2 @ C @ Q5 @ T3 )
     => ( ( entails @ P @ P2 )
       => ( ! [X3: nat] : ( entails @ ( Q5 @ X3 ) @ ( Q @ X3 ) )
         => ( ( ord_less_eq_nat @ T3 @ T )
           => ( time_htt_nat @ P @ C @ Q @ T ) ) ) ) ) ).

% htt_cons_rule
thf(fact_5964_nth__rule,axiom,
    ! [I: nat,Xs: list_int,A: array_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( hoare_3065115510600077593le_int @ ( snga_assn_int @ A @ Xs ) @ ( array_nth_int @ A @ I )
        @ ^ [R5: int] :
            ( times_times_assn @ ( snga_assn_int @ A @ Xs )
            @ ( pure_assn
              @ ( R5
                = ( nth_int @ Xs @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_5965_nth__rule,axiom,
    ! [I: nat,Xs: list_o,A: array_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( hoare_hoare_triple_o @ ( snga_assn_o @ A @ Xs ) @ ( array_nth_o @ A @ I )
        @ ^ [R5: $o] :
            ( times_times_assn @ ( snga_assn_o @ A @ Xs )
            @ ( pure_assn
              @ ( R5
                = ( nth_o @ Xs @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_5966_nth__rule,axiom,
    ! [I: nat,Xs: list_option_nat,A: array_option_nat] :
      ( ( ord_less_nat @ I @ ( size_s6086282163384603972on_nat @ Xs ) )
     => ( hoare_7629718768684598413on_nat @ ( snga_assn_option_nat @ A @ Xs ) @ ( array_nth_option_nat @ A @ I )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( snga_assn_option_nat @ A @ Xs )
            @ ( pure_assn
              @ ( R5
                = ( nth_option_nat @ Xs @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_5967_nth__rule,axiom,
    ! [I: nat,Xs: list_nat,A: array_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( hoare_3067605981109127869le_nat @ ( snga_assn_nat @ A @ Xs ) @ ( array_nth_nat @ A @ I )
        @ ^ [R5: nat] :
            ( times_times_assn @ ( snga_assn_nat @ A @ Xs )
            @ ( pure_assn
              @ ( R5
                = ( nth_nat @ Xs @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_5968_nth__rule,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,A: array_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs ) )
     => ( hoare_1429296392585015714_VEBTi @ ( snga_assn_VEBT_VEBTi @ A @ Xs ) @ ( array_nth_VEBT_VEBTi @ A @ I )
        @ ^ [R5: vEBT_VEBTi] :
            ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ A @ Xs )
            @ ( pure_assn
              @ ( R5
                = ( nth_VEBT_VEBTi @ Xs @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_5969_return__cons__rule,axiom,
    ! [P: assn,Q: $o > assn,X: $o] :
      ( ( entails @ P @ ( Q @ X ) )
     => ( hoare_hoare_triple_o @ P @ ( heap_Time_return_o @ X ) @ Q ) ) ).

% return_cons_rule
thf(fact_5970_return__cons__rule,axiom,
    ! [P: assn,Q: option_nat > assn,X: option_nat] :
      ( ( entails @ P @ ( Q @ X ) )
     => ( hoare_7629718768684598413on_nat @ P @ ( heap_T3487192422709364219on_nat @ X ) @ Q ) ) ).

% return_cons_rule
thf(fact_5971_return__cons__rule,axiom,
    ! [P: assn,Q: nat > assn,X: nat] :
      ( ( entails @ P @ ( Q @ X ) )
     => ( hoare_3067605981109127869le_nat @ P @ ( heap_Time_return_nat @ X ) @ Q ) ) ).

% return_cons_rule
thf(fact_5972_return__cons__rule,axiom,
    ! [P: assn,Q: vEBT_VEBTi > assn,X: vEBT_VEBTi] :
      ( ( entails @ P @ ( Q @ X ) )
     => ( hoare_1429296392585015714_VEBTi @ P @ ( heap_T3630416162098727440_VEBTi @ X ) @ Q ) ) ).

% return_cons_rule
thf(fact_5973_fi__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,Ps: assn,F2: assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F2 ) )
       => ( hoare_hoare_triple_o @ Ps @ C
          @ ^ [X2: $o] : ( times_times_assn @ ( Q @ X2 ) @ F2 ) ) ) ) ).

% fi_rule
thf(fact_5974_fi__rule,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,Ps: assn,F2: assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F2 ) )
       => ( hoare_7629718768684598413on_nat @ Ps @ C
          @ ^ [X2: option_nat] : ( times_times_assn @ ( Q @ X2 ) @ F2 ) ) ) ) ).

% fi_rule
thf(fact_5975_fi__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,Ps: assn,F2: assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F2 ) )
       => ( hoare_3067605981109127869le_nat @ Ps @ C
          @ ^ [X2: nat] : ( times_times_assn @ ( Q @ X2 ) @ F2 ) ) ) ) ).

% fi_rule
thf(fact_5976_fi__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,Ps: assn,F2: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F2 ) )
       => ( hoare_1429296392585015714_VEBTi @ Ps @ C
          @ ^ [X2: vEBT_VEBTi] : ( times_times_assn @ ( Q @ X2 ) @ F2 ) ) ) ) ).

% fi_rule
thf(fact_5977_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: 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 @ Q2 @ 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 @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_5978_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ( ln_ln_real @ X )
            = ( ln_ln_real @ Y ) )
          = ( X = Y ) ) ) ) ).

% ln_inj_iff
thf(fact_5979_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% ln_less_cancel_iff
thf(fact_5980_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_5981_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_5982_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_5983_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_5984_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_5985_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_5986_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_5987_unique__quotient,axiom,
    ! [A: int,B: int,Q2: int,R2: int,Q6: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R4 ) )
       => ( Q2 = Q6 ) ) ) ).

% unique_quotient
thf(fact_5988_unique__remainder,axiom,
    ! [A: int,B: int,Q2: int,R2: int,Q6: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R4 ) )
       => ( R2 = R4 ) ) ) ).

% unique_remainder
thf(fact_5989_eucl__rel__int__by0,axiom,
    ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).

% eucl_rel_int_by0
thf(fact_5990_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( modulo_modulo_int @ K @ L )
        = R2 ) ) ).

% mod_int_unique
thf(fact_5991_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( divide_divide_int @ K @ L )
        = Q2 ) ) ).

% div_int_unique
thf(fact_5992_ln__less__self,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).

% ln_less_self
thf(fact_5993_log__def,axiom,
    ( log
    = ( ^ [A4: real,X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ A4 ) ) ) ) ).

% log_def
thf(fact_5994_ln__bound,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).

% ln_bound
thf(fact_5995_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_5996_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_5997_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_5998_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_5999_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q2 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_6000_eucl__rel__int,axiom,
    ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).

% eucl_rel_int
thf(fact_6001_assn__aci_I10_J,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
      = ( times_times_assn @ ( times_times_assn @ A @ C ) @ B ) ) ).

% assn_aci(10)
thf(fact_6002_star__aci_I3_J,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ A @ ( times_times_assn @ B @ C ) )
      = ( times_times_assn @ B @ ( times_times_assn @ A @ C ) ) ) ).

% star_aci(3)
thf(fact_6003_star__aci_I2_J,axiom,
    ( times_times_assn
    = ( ^ [A4: assn,B2: assn] : ( times_times_assn @ B2 @ A4 ) ) ) ).

% star_aci(2)
thf(fact_6004_star__assoc,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).

% star_assoc
thf(fact_6005_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entails @ P @ Q ) ) ).

% is_entails
thf(fact_6006_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_6007_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_6008_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_6009_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_6010_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_6011_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_6012_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_6013_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_6014_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_6015_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_6016_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q2 ) @ 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 )
             => ( Q2 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_6017_fr__rot,axiom,
    ! [A2: assn,B3: assn,C5: assn] :
      ( ( entails @ ( times_times_assn @ A2 @ B3 ) @ C5 )
     => ( entails @ ( times_times_assn @ B3 @ A2 ) @ C5 ) ) ).

% fr_rot
thf(fact_6018_fr__refl,axiom,
    ! [A2: assn,B3: assn,C5: assn] :
      ( ( entails @ A2 @ B3 )
     => ( entails @ ( times_times_assn @ A2 @ C5 ) @ ( times_times_assn @ B3 @ C5 ) ) ) ).

% fr_refl
thf(fact_6019_fr__rot__rhs,axiom,
    ! [A2: assn,B3: assn,C5: assn] :
      ( ( entails @ A2 @ ( times_times_assn @ B3 @ C5 ) )
     => ( entails @ A2 @ ( times_times_assn @ C5 @ B3 ) ) ) ).

% fr_rot_rhs
thf(fact_6020_ent__frame__fwd,axiom,
    ! [P: assn,R3: assn,Ps: assn,F2: assn,Q: assn] :
      ( ( entails @ P @ R3 )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F2 ) )
       => ( ( entails @ ( times_times_assn @ R3 @ F2 ) @ Q )
         => ( entails @ Ps @ Q ) ) ) ) ).

% ent_frame_fwd
thf(fact_6021_norm__assertion__simps_I2_J,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% norm_assertion_simps(2)
thf(fact_6022_norm__assertion__simps_I1_J,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% norm_assertion_simps(1)
thf(fact_6023_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_6024_frame__rule__left,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,R3: assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ R3 @ P ) @ C
        @ ^ [X2: $o] : ( times_times_assn @ R3 @ ( Q @ X2 ) ) ) ) ).

% frame_rule_left
thf(fact_6025_frame__rule__left,axiom,
    ! [P: assn,C: heap_T2636463487746394924on_nat,Q: option_nat > assn,R3: assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C @ Q )
     => ( hoare_7629718768684598413on_nat @ ( times_times_assn @ R3 @ P ) @ C
        @ ^ [X2: option_nat] : ( times_times_assn @ R3 @ ( Q @ X2 ) ) ) ) ).

% frame_rule_left
thf(fact_6026_frame__rule__left,axiom,
    ! [P: assn,C: heap_Time_Heap_nat,Q: nat > assn,R3: assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C @ Q )
     => ( hoare_3067605981109127869le_nat @ ( times_times_assn @ R3 @ P ) @ C
        @ ^ [X2: nat] : ( times_times_assn @ R3 @ ( Q @ X2 ) ) ) ) ).

% frame_rule_left
thf(fact_6027_frame__rule__left,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,R3: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ R3 @ P ) @ C
        @ ^ [X2: vEBT_VEBTi] : ( times_times_assn @ R3 @ ( Q @ X2 ) ) ) ) ).

% frame_rule_left
thf(fact_6028_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ 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 @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_6029_mod__frame__fwd,axiom,
    ! [Ps: assn,H2: produc3658429121746597890et_nat,P: assn,R3: assn,F2: assn] :
      ( ( rep_assn @ Ps @ H2 )
     => ( ( entails @ P @ R3 )
       => ( ( entails @ Ps @ ( times_times_assn @ P @ F2 ) )
         => ( rep_assn @ ( times_times_assn @ R3 @ F2 ) @ H2 ) ) ) ) ).

% mod_frame_fwd
thf(fact_6030_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q @ T )
      = ( B
       => ( time_htt_VEBT_VEBTi @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_6031_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( time_htt_option_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q @ T )
      = ( B
       => ( time_htt_option_nat @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_6032_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B: $o,F: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( time_htt_nat @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ F @ Q @ T )
      = ( B
       => ( time_htt_nat @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_6033_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ ( times_times_assn @ ( pure_assn @ B ) @ P ) @ F @ Q @ T )
      = ( B
       => ( time_htt_VEBT_VEBTi @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_6034_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( time_htt_option_nat @ ( times_times_assn @ ( pure_assn @ B ) @ P ) @ F @ Q @ T )
      = ( B
       => ( time_htt_option_nat @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_6035_norm__pre__pure__iff__htt_H,axiom,
    ! [B: $o,P: assn,F: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( time_htt_nat @ ( times_times_assn @ ( pure_assn @ B ) @ P ) @ F @ Q @ T )
      = ( B
       => ( time_htt_nat @ P @ F @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_6036_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R2 ) )
      = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_6037_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) @ N )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6038_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s7982070591426661849_VEBTi @ Ys ) ) )
     => ( ( nth_Pr316670251186196177_VEBTi @ ( produc316462671093861988_VEBTi @ Xs @ Ys ) @ N )
        = ( produc6084888613844515218_VEBTi @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) @ ( nth_VEBT_VEBTi @ Ys @ ( modulo_modulo_nat @ N @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6039_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr8725177398587324397T_VEBT @ ( produc1285381384045549624T_VEBT @ Xs @ Ys ) @ N )
        = ( produc7053807326796202854T_VEBT @ ( nth_VEBT_VEBTi @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6040_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) @ ( size_s7982070591426661849_VEBTi @ Ys ) ) )
     => ( ( nth_Pr6329974346453275474_VEBTi @ ( produc194614972289024177_VEBTi @ Xs @ Ys ) @ N )
        = ( produc436343169921013763_VEBTi @ ( nth_VEBT_VEBTi @ Xs @ ( divide_divide_nat @ N @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) @ ( nth_VEBT_VEBTi @ Ys @ ( modulo_modulo_nat @ N @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6041_product__nth,axiom,
    ! [N: nat,Xs: list_num,Ys: list_num] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_num @ Xs ) @ ( size_size_list_num @ Ys ) ) )
     => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs @ Ys ) @ N )
        = ( product_Pair_num_num @ ( nth_num @ Xs @ ( 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_6042_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_real] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_real @ Ys ) ) )
     => ( ( nth_Pr6842391030413306568T_real @ ( produc4908677263432625371T_real @ Xs @ Ys ) @ N )
        = ( produc8117437818029410057T_real @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_real @ Ys ) ) ) @ ( nth_real @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_real @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6043_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Ys: list_real] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) @ ( size_size_list_real @ Ys ) ) )
     => ( ( nth_Pr3433448822664029129i_real @ ( produc5476717833281694120i_real @ Xs @ Ys ) @ N )
        = ( produc8457151488442208762i_real @ ( nth_VEBT_VEBTi @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_real @ Ys ) ) ) @ ( nth_real @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_real @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6044_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) @ N )
        = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( 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_6045_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Ys: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr3306050735993963089EBTi_o @ ( product_VEBT_VEBTi_o @ Xs @ Ys ) @ N )
        = ( produc8194178580519725514EBTi_o @ ( nth_VEBT_VEBTi @ Xs @ ( 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_6046_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) @ N )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( 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_6047_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_6048_vebt__maxti_Oelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_maxti @ X )
        = Y )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leafi @ A6 @ B5 ) )
           => ~ ( ( B5
                 => ( Y
                    = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                & ( ~ B5
                 => ( ( A6
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                    & ( ~ A6
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( X
                = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y
             != ( heap_T3487192422709364219on_nat @ none_nat ) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
                    ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
               => ( Y
                 != ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma2 ) ) ) ) ) ) ) ).

% vebt_maxti.elims
thf(fact_6049_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_6050_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_6051_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = ( uminus1482373934393186551omplex @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_6052_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_6053_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_6054_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_6055_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_6056_add_Oinverse__inverse,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_6057_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_6058_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_6059_VEBTi_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
      ( ( ( vEBT_Leafi @ X21 @ X222 )
        = ( vEBT_Leafi @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% VEBTi.inject(2)
thf(fact_6060_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_6061_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_6062_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_6063_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_6064_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_6065_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_6066_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_6067_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_6068_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_6069_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_6070_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_6071_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_6072_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_6073_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_6074_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_6075_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_6076_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_6077_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_6078_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_6079_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_6080_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_6081_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_6082_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_6083_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_6084_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_6085_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_6086_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_6087_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_6088_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_6089_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_6090_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_6091_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_6092_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_6093_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_6094_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_6095_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_6096_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_6097_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_6098_add__minus__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_6099_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_6100_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_6101_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_6102_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_6103_minus__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_6104_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_6105_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_6106_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_6107_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_6108_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_6109_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_6110_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_6111_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_6112_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_6113_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_6114_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_6115_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_6116_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_6117_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_6118_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_6119_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_6120_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_6121_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_6122_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_6123_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_6124_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_6125_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_6126_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_6127_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_6128_minus__diff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
      = ( minus_minus_complex @ B @ A ) ) ).

% minus_diff_eq
thf(fact_6129_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% minus_diff_eq
thf(fact_6130_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_6131_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_6132_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ A @ B ) ) ).

% div_minus_minus
thf(fact_6133_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_6134_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_6135_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_6136_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_6137_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_6138_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_6139_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_6140_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y ) )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_6141_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_6142_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_6143_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_6144_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_6145_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_6146_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_6147_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_6148_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_6149_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_6150_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_6151_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_6152_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_6153_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_6154_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_6155_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_6156_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_6157_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_6158_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_6159_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_6160_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_6161_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_6162_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_6163_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_6164_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_6165_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_6166_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_6167_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_6168_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_6169_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_6170_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_6171_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_6172_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_6173_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_6174_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_6175_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_6176_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_6177_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_6178_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_6179_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_6180_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_6181_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_6182_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_6183_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_6184_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_6185_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_6186_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_6187_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_6188_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_6189_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_6190_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_6191_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_6192_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_6193_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_6194_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_6195_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_6196_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_6197_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6198_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6199_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6200_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6201_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6202_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_6203_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_6204_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_6205_mult__minus1,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1
thf(fact_6206_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_6207_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_6208_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_6209_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_6210_mult__minus1__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1_right
thf(fact_6211_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_6212_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_6213_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_6214_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_6215_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_6216_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_6217_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_6218_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_6219_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_6220_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_6221_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_6222_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_6223_div__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% div_minus1_right
thf(fact_6224_divide__minus1,axiom,
    ! [X: real] :
      ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X ) ) ).

% divide_minus1
thf(fact_6225_divide__minus1,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% divide_minus1
thf(fact_6226_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_6227_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_6228_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_6229_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_6230_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_6231_of__int__minus,axiom,
    ! [Z: int] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
      = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_minus
thf(fact_6232_of__int__minus,axiom,
    ! [Z: int] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z ) )
      = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_minus
thf(fact_6233_of__int__minus,axiom,
    ! [Z: int] :
      ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ Z ) )
      = ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_int_minus
thf(fact_6234_of__int__minus,axiom,
    ! [Z: int] :
      ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z ) )
      = ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z ) ) ) ).

% of_int_minus
thf(fact_6235_of__int__minus,axiom,
    ! [Z: int] :
      ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z ) )
      = ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_minus
thf(fact_6236_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_6237_negative__zle,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_6238_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_6239_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_6240_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_6241_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_6242_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_6243_length__product,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_s3932428310213730859l_real @ ( product_real_real @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_6244_length__product,axiom,
    ! [Xs: list_real,Ys: list_o] :
      ( ( size_s987546567493390085real_o @ ( product_real_o @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_6245_length__product,axiom,
    ! [Xs: list_real,Ys: list_nat] :
      ( ( size_s1877336372972134351al_nat @ ( product_real_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_6246_length__product,axiom,
    ! [Xs: list_real,Ys: list_int] :
      ( ( size_s8610625264895183403al_int @ ( product_real_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_6247_length__product,axiom,
    ! [Xs: list_o,Ys: list_real] :
      ( ( size_s2624279037499656343o_real @ ( product_o_real @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_6248_length__product,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_6249_length__product,axiom,
    ! [Xs: list_o,Ys: list_nat] :
      ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_6250_length__product,axiom,
    ! [Xs: list_o,Ys: list_int] :
      ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_6251_length__product,axiom,
    ! [Xs: list_nat,Ys: list_real] :
      ( ( size_s7910714270633306959t_real @ ( product_nat_real @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_6252_length__product,axiom,
    ! [Xs: list_nat,Ys: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_6253_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_6254_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_6255_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_6256_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_6257_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_6258_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_6259_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_6260_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_6261_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_6262_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_6263_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_6264_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_6265_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_6266_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_6267_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_6268_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_6269_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_6270_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_6271_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_6272_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_6273_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_6274_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_6275_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_6276_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_6277_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_6278_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_6279_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_6280_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_6281_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_6282_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_6283_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_6284_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_6285_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_6286_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_6287_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_6288_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_6289_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_6290_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_6291_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_6292_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_6293_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_6294_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_6295_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_6296_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_6297_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_6298_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_6299_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_6300_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_6301_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_6302_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_6303_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_6304_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_6305_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_6306_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_6307_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_6308_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_6309_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6310_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6311_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6312_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6313_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6314_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6315_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6316_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6317_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6318_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6319_negative__zless,axiom,
    ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_6320_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6321_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6322_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6323_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6324_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6325_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6326_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6327_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6328_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6329_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6330_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6331_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6332_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6333_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6334_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6335_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_6336_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_6337_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_6338_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_6339_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_6340_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_6341_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_6342_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_6343_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_6344_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_6345_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_6346_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_6347_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_6348_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_6349_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_6350_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_6351_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_6352_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_6353_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_6354_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_6355_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_6356_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_6357_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_6358_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_6359_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_6360_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_6361_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_6362_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_6363_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_6364_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_6365_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_6366_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_6367_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_6368_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_6369_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_6370_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_6371_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_6372_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_6373_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_6374_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_6375_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_6376_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_6377_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_6378_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_6379_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_6380_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_6381_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_6382_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_6383_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_6384_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_6385_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_6386_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_6387_int__div__minus__is__minus1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ( divide_divide_int @ A @ B )
          = ( uminus_uminus_int @ A ) )
        = ( B
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% int_div_minus_is_minus1
thf(fact_6388_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_6389_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_6390_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_6391_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_6392_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_6393_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_6394_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_6395_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_6396_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_6397_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_6398_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_6399_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_6400_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_6401_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_6402_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_6403_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_6404_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_6405_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_6406_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_6407_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_6408_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_6409_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_6410_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_6411_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_6412_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_6413_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_6414_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_6415_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_6416_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_6417_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_6418_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_6419_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_6420_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_6421_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_6422_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_6423_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_6424_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_6425_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_6426_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_6427_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_6428_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_6429_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_6430_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_6431_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_6432_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_6433_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_6434_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_6435_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_6436_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_6437_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_6438_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_6439_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_6440_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% numeral_div_minus_numeral
thf(fact_6441_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% minus_numeral_div_numeral
thf(fact_6442_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_6443_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_6444_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_6445_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_6446_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_6447_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_6448_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_6449_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_6450_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_6451_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_6452_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_6453_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_6454_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_6455_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_6456_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_6457_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_6458_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_6459_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_6460_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_6461_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_6462_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_6463_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_6464_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_6465_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_6466_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_6467_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_6468_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_6469_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_6470_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_6471_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_6472_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_6473_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_6474_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_6475_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_6476_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_6477_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_6478_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_6479_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_6480_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_6481_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_6482_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_6483_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_6484_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_6485_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_6486_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_6487_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_6488_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_6489_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_6490_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_6491_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_6492_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_6493_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_6494_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_6495_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_6496_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_6497_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_6498_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_6499_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_6500_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_6501_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_6502_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6503_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6504_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( uminus1482373934393186551omplex @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6505_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( uminus1351360451143612070nteger @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6506_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6507_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_6508_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_6509_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% equation_minus_iff
thf(fact_6510_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_6511_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_6512_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_6513_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_6514_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_6515_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_6516_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_6517_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_6518_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_6519_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_6520_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_6521_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_6522_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_6523_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_6524_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_6525_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_6526_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_6527_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_6528_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_6529_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_6530_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_6531_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_6532_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_6533_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_6534_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_6535_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_6536_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_int @ M )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6537_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_real @ M )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6538_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numera6690914467698888265omplex @ M )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6539_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numera6620942414471956472nteger @ M )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6540_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_rat @ M )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6541_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
     != ( numeral_numeral_int @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6542_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
     != ( numeral_numeral_real @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6543_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
     != ( numera6690914467698888265omplex @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6544_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
     != ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6545_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
     != ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6546_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( uminus_uminus_int @ A2 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_6547_group__cancel_Oneg1,axiom,
    ! [A2: real,K: real,A: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( uminus_uminus_real @ A2 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_6548_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2
        = ( plus_plus_complex @ K @ A ) )
     => ( ( uminus1482373934393186551omplex @ A2 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_6549_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2
        = ( plus_p5714425477246183910nteger @ K @ A ) )
     => ( ( uminus1351360451143612070nteger @ A2 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_6550_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( uminus_uminus_rat @ A2 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_6551_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_6552_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_6553_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_6554_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_6555_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_6556_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_6557_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_6558_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_6559_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_6560_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_6561_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_6562_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_6563_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_6564_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_6565_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_6566_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_6567_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_6568_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_6569_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_6570_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_6571_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_6572_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_6573_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_6574_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_6575_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_6576_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_6577_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_6578_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_6579_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_6580_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_6581_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_6582_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_6583_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_6584_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_6585_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_6586_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_6587_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% div_minus_right
thf(fact_6588_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_6589_minus__divide__left,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_6590_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_6591_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_6592_minus__divide__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( divide1717551699836669952omplex @ A @ B ) ) ).

% minus_divide_divide
thf(fact_6593_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_6594_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_6595_minus__divide__right,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_divide_right
thf(fact_6596_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_6597_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_6598_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_6599_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: int,B: int,A5: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = ( modulo_modulo_int @ A5 @ B ) )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
        = ( modulo_modulo_int @ ( uminus_uminus_int @ A5 ) @ B ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
thf(fact_6600_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A5: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = ( modulo364778990260209775nteger @ A5 @ B ) )
     => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A5 ) @ B ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
thf(fact_6601_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_6602_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_6603_uminus__int__code_I1_J,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% uminus_int_code(1)
thf(fact_6604_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N2: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% int_cases2
thf(fact_6605_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).

% signed_take_bit_minus
thf(fact_6606_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( size_size_VEBT_VEBTi @ ( vEBT_Leafi @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBTi.size(4)
thf(fact_6607_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_6608_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_6609_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_6610_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_6611_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_6612_VEBTi_Oexhaust,axiom,
    ! [Y: vEBT_VEBTi] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: array_VEBT_VEBTi,X142: vEBT_VEBTi] :
          ( Y
         != ( vEBT_Nodei @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X223: $o] :
            ( Y
           != ( vEBT_Leafi @ X212 @ X223 ) ) ) ).

% VEBTi.exhaust
thf(fact_6613_VEBTi_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,X21: $o,X222: $o] :
      ( ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leafi @ X21 @ X222 ) ) ).

% VEBTi.distinct(1)
thf(fact_6614_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6615_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6616_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6617_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6618_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6619_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6620_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6621_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6622_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6623_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_6624_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_6625_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_6626_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_6627_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6628_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6629_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6630_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6631_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_6632_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_6633_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_6634_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_6635_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_6636_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_6637_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_6638_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_6639_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_6640_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_6641_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_6642_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_6643_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_6644_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_6645_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_6646_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_6647_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_6648_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_6649_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_6650_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_6651_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_6652_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_6653_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_6654_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_6655_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_6656_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_6657_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_6658_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_6659_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_6660_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_6661_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_6662_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_6663_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_6664_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_6665_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_6666_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_6667_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_6668_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_6669_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_6670_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_6671_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_6672_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_6673_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_6674_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_6675_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_6676_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_6677_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_6678_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_6679_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_6680_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_6681_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6682_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6683_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6684_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6685_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6686_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ N )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_6687_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ N )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_6688_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ N )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_6689_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ N )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_6690_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ N )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_6691_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_6692_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_6693_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_6694_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_6695_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_6696_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_6697_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_6698_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_6699_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_6700_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_6701_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_6702_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_6703_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_6704_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_6705_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_6706_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_6707_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A4: int,B2: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6708_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A4: real,B2: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6709_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A4: complex,B2: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6710_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A4: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6711_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A4: rat,B2: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6712_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A4: int,B2: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6713_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A4: real,B2: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6714_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A4: complex,B2: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6715_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A4: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6716_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A4: rat,B2: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6717_group__cancel_Osub2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B3 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6718_group__cancel_Osub2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3
        = ( plus_plus_real @ K @ B ) )
     => ( ( minus_minus_real @ A @ B3 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6719_group__cancel_Osub2,axiom,
    ! [B3: complex,K: complex,B: complex,A: complex] :
      ( ( B3
        = ( plus_plus_complex @ K @ B ) )
     => ( ( minus_minus_complex @ A @ B3 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6720_group__cancel_Osub2,axiom,
    ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B3
        = ( plus_p5714425477246183910nteger @ K @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B3 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6721_group__cancel_Osub2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B3 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6722_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_6723_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_6724_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_6725_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_6726_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_6727_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_6728_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_6729_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_6730_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_6731_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_6732_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_6733_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_6734_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ( times_times_int @ M @ N )
        = one_one_int )
      = ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_6735_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( ( times_times_int @ M @ N )
        = one_one_int )
     => ( ( M = one_one_int )
        | ( M
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_6736_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_6737_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
thf(fact_6738_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
thf(fact_6739_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_6740_vebt__buildupi_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildupi @ zero_zero_nat )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% vebt_buildupi.simps(1)
thf(fact_6741_VEBT__internal_Ovebt__buildupi_H_Osimps_I1_J,axiom,
    ( ( vEBT_V739175172307565963ildupi @ zero_zero_nat )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(1)
thf(fact_6742_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_6743_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_6744_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_6745_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_6746_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_6747_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_6748_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_6749_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_6750_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_6751_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_6752_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_6753_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_6754_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_6755_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_6756_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_6757_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_6758_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_6759_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_6760_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_6761_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_6762_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_6763_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_6764_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_6765_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_6766_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_6767_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_6768_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_6769_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_6770_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_6771_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_6772_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_6773_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_6774_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_6775_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_6776_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_6777_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_6778_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_6779_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_le_neg_one
thf(fact_6780_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_le_neg_one
thf(fact_6781_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_le_neg_one
thf(fact_6782_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% neg_numeral_le_neg_one
thf(fact_6783_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% neg_numeral_le_neg_one
thf(fact_6784_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% neg_numeral_le_neg_one
thf(fact_6785_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% neg_numeral_le_neg_one
thf(fact_6786_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_6787_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_le_numeral
thf(fact_6788_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_le_numeral
thf(fact_6789_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_le_numeral
thf(fact_6790_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_6791_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_le_one
thf(fact_6792_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_le_one
thf(fact_6793_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_le_one
thf(fact_6794_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_6795_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_6796_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_6797_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_6798_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_6799_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_6800_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_6801_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_6802_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_less_neg_one
thf(fact_6803_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_less_neg_one
thf(fact_6804_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_6805_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_less_neg_one
thf(fact_6806_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_less_numeral
thf(fact_6807_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_less_numeral
thf(fact_6808_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_6809_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_less_numeral
thf(fact_6810_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_less_one
thf(fact_6811_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_less_one
thf(fact_6812_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_6813_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_less_one
thf(fact_6814_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_6815_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_6816_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_6817_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_6818_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_6819_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_6820_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_6821_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_6822_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_6823_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_6824_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_6825_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_6826_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_6827_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_6828_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_6829_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_6830_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_6831_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_6832_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_6833_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_6834_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_6835_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_6836_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_6837_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_6838_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_6839_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_6840_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_6841_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_6842_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_6843_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_6844_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_6845_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_6846_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_6847_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_6848_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_6849_power__minus__Bit0,axiom,
    ! [X: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6850_power__minus__Bit0,axiom,
    ! [X: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6851_power__minus__Bit0,axiom,
    ! [X: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6852_power__minus__Bit0,axiom,
    ! [X: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6853_power__minus__Bit0,axiom,
    ! [X: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6854_power__minus__Bit1,axiom,
    ! [X: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6855_power__minus__Bit1,axiom,
    ! [X: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6856_power__minus__Bit1,axiom,
    ! [X: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6857_power__minus__Bit1,axiom,
    ! [X: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6858_power__minus__Bit1,axiom,
    ! [X: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6859_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_6860_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_6861_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% int_cases4
thf(fact_6862_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_6863_negative__zle__0,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_6864_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% nonpos_int_cases
thf(fact_6865_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_6866_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_6867_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
    ( ( vEBT_V739175172307565963ildupi @ ( suc @ zero_zero_nat ) )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(2)
thf(fact_6868_vebt__buildupi_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildupi @ ( suc @ zero_zero_nat ) )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% vebt_buildupi.simps(2)
thf(fact_6869_vebt__assn__raw_Ocases,axiom,
    ! [X: produc3625547720036274456_VEBTi] :
      ( ! [A6: $o,B5: $o,Ai: $o,Bi: $o] :
          ( X
         != ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ A6 @ B5 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) )
     => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT,Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
            ( X
           != ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) ) )
       => ( ! [V2: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] :
              ( X
             != ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ V2 @ Va @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) )
         => ~ ! [Vd3: $o,Ve3: $o,V2: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                ( X
               != ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V2 @ Va @ Vb3 @ Vc3 ) ) ) ) ) ) ).

% vebt_assn_raw.cases
thf(fact_6870_vebt__assn__raw_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Ai2: $o,Bi2: $o] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ A @ B ) @ ( vEBT_Leafi @ Ai2 @ Bi2 ) )
      = ( pure_assn
        @ ( ( Ai2 = A )
          & ( Bi2 = B ) ) ) ) ).

% vebt_assn_raw.simps(1)
thf(fact_6871_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_6872_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_6873_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_6874_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_6875_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_6876_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_6877_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_6878_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_6879_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_6880_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_6881_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_6882_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_6883_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_6884_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_6885_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_6886_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_6887_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_6888_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_6889_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_6890_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_6891_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_6892_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_6893_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_6894_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_6895_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_6896_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_6897_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_6898_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_6899_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_6900_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_6901_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_6902_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_6903_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_6904_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_6905_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_6906_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_6907_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_6908_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_6909_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_6910_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_6911_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
       => ~ ! [N2: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% int_cases3
thf(fact_6912_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_6913_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_6914_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N2: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% negD
thf(fact_6915_verit__less__mono__div__int2,axiom,
    ! [A2: int,B3: int,N: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_6916_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_6917_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_6918_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_6919_vebt__minti_Ocases,axiom,
    ! [X: vEBT_VEBTi] :
      ( ! [A6: $o,B5: $o] :
          ( X
         != ( vEBT_Leafi @ A6 @ B5 ) )
     => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
            ( X
           != ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
              ( X
             != ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ).

% vebt_minti.cases
thf(fact_6920_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_6921_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_6922_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_6923_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_6924_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_6925_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_6926_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_6927_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_6928_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_6929_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_6930_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_6931_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_6932_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_6933_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_6934_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_6935_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_6936_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_6937_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_6938_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_6939_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_6940_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_6941_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_6942_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_6943_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_6944_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_6945_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_6946_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6947_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6948_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6949_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6950_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6951_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% neg_int_cases
thf(fact_6952_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
        = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).

% minus_mod_int_eq
thf(fact_6953_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_6954_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_6955_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_6956_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ 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 @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_6957_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_6958_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_6959_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_6960_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_6961_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_6962_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_6963_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_6964_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_6965_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_6966_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_6967_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_6968_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_6969_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_6970_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_6971_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_6972_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_6973_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_6974_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_6975_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_6976_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_6977_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_6978_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_6979_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( divide_divide_int @ K @ L )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_6980_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_6981_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_6982_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_6983_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_6984_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_6985_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K2: int] :
              ( ( P @ K2 )
             => ( ( K2 != zero_zero_int )
               => ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K2: int] :
                ( ( P @ K2 )
               => ( ( K2
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_6986_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_6987_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_ri631733984087533419it_int @ N @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_6988_m1mod2k,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ).

% m1mod2k
thf(fact_6989_vebt__minti_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( A
       => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A @ B ) )
          = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
      & ( ~ A
       => ( ( B
           => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A @ B ) )
              = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
          & ( ~ B
           => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A @ B ) )
              = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) ).

% vebt_minti.simps(1)
thf(fact_6990_vebt__maxti_Osimps_I1_J,axiom,
    ! [B: $o,A: $o] :
      ( ( B
       => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A @ B ) )
          = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A @ B ) )
              = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A @ B ) )
              = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) ).

% vebt_maxti.simps(1)
thf(fact_6991_sb__dec__lem_H,axiom,
    ! [K: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ A ) ) ) ).

% sb_dec_lem'
thf(fact_6992_m1mod22k,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ one_one_int ) ) ).

% m1mod22k
thf(fact_6993_sb__inc__lem_H,axiom,
    ! [A: int,K: nat] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) ) ) ).

% sb_inc_lem'
thf(fact_6994_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_6995_sb__dec__lem,axiom,
    ! [K: nat,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ A ) )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ ( plus_plus_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ A ) ) ) ).

% sb_dec_lem
thf(fact_6996_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N4: nat,A4: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_6997_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,A4: int] : ( if_int @ ( N4 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_6998_vebt__minti_Oelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_minti @ X )
        = Y )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leafi @ A6 @ B5 ) )
           => ~ ( ( A6
                 => ( Y
                    = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                & ( ~ A6
                 => ( ( B5
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                    & ( ~ B5
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( X
                = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y
             != ( heap_T3487192422709364219on_nat @ none_nat ) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
                    ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
               => ( Y
                 != ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi2 ) ) ) ) ) ) ) ).

% vebt_minti.elims
thf(fact_6999_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_7000_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_7001_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_7002_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_7003_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_7004_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_7005_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_7006_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_7007_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_7008_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_7009_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_minus_bit1
thf(fact_7010_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L3: 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 @ L3 @ ( 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 ) ) @ L3 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_7011_and__int_Oelims,axiom,
    ! [X: int,Xa3: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa3 )
        = Y )
     => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa3 @ ( 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 ) ) @ Xa3 ) ) ) ) ) )
        & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa3 @ ( 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 ) ) @ Xa3 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_7012_and_Oidem,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ A )
      = A ) ).

% and.idem
thf(fact_7013_and_Oidem,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ A )
      = A ) ).

% and.idem
thf(fact_7014_and_Oleft__idem,axiom,
    ! [A: int,B: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( bit_se725231765392027082nd_int @ A @ B ) ) ).

% and.left_idem
thf(fact_7015_and_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).

% and.left_idem
thf(fact_7016_and_Oright__idem,axiom,
    ! [A: int,B: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ B )
      = ( bit_se725231765392027082nd_int @ A @ B ) ) ).

% and.right_idem
thf(fact_7017_and_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ B )
      = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).

% and.right_idem
thf(fact_7018_and__zero__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% and_zero_eq
thf(fact_7019_and__zero__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% and_zero_eq
thf(fact_7020_and__zero__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% and_zero_eq
thf(fact_7021_zero__and__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% zero_and_eq
thf(fact_7022_zero__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% zero_and_eq
thf(fact_7023_zero__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_and_eq
thf(fact_7024_bit_Oconj__zero__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ zero_z3403309356797280102nteger @ X )
      = zero_z3403309356797280102nteger ) ).

% bit.conj_zero_left
thf(fact_7025_bit_Oconj__zero__left,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
      = zero_zero_int ) ).

% bit.conj_zero_left
thf(fact_7026_bit_Oconj__zero__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ X @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% bit.conj_zero_right
thf(fact_7027_bit_Oconj__zero__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
      = zero_zero_int ) ).

% bit.conj_zero_right
thf(fact_7028_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_7029_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_7030_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_7031_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_7032_and_Oleft__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_7033_and_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_7034_and_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = A ) ).

% and.right_neutral
thf(fact_7035_and_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = A ) ).

% and.right_neutral
thf(fact_7036_bit_Oconj__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = X ) ).

% bit.conj_one_right
thf(fact_7037_bit_Oconj__one__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = X ) ).

% bit.conj_one_right
thf(fact_7038_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_7039_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ( suc @ N )
        = ( numeral_numeral_nat @ K ) )
      = ( N
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_7040_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N ) )
      = ( ( pred_numeral @ K )
        = N ) ) ).

% eq_numeral_Suc
thf(fact_7041_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% and_nonnegative_int_iff
thf(fact_7042_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        & ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_7043_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = one_one_int ) ).

% and_numerals(8)
thf(fact_7044_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = one_one_nat ) ).

% and_numerals(8)
thf(fact_7045_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = one_one_int ) ).

% and_numerals(2)
thf(fact_7046_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_numerals(2)
thf(fact_7047_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_7048_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_7049_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).

% less_numeral_Suc
thf(fact_7050_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_7051_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).

% le_numeral_Suc
thf(fact_7052_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_7053_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).

% diff_numeral_Suc
thf(fact_7054_max__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_7055_max__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).

% max_numeral_Suc
thf(fact_7056_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_7057_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_7058_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_7059_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_7060_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_7061_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% and_numerals(5)
thf(fact_7062_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_7063_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_7064_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = zero_z3403309356797280102nteger ) ).

% and_numerals(1)
thf(fact_7065_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = zero_zero_int ) ).

% and_numerals(1)
thf(fact_7066_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_numerals(1)
thf(fact_7067_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_7068_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_7069_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_7070_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_7071_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_7072_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_7073_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_7074_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_7075_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_7076_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_7077_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_7078_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_bit0
thf(fact_7079_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_7080_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_7081_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_minus_bit0
thf(fact_7082_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_7083_of__int__and__eq,axiom,
    ! [K: int,L: int] :
      ( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).

% of_int_and_eq
thf(fact_7084_of__nat__and__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N ) )
      = ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_and_eq
thf(fact_7085_of__nat__and__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N ) )
      = ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_and_eq
thf(fact_7086_and_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ C )
      = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).

% and.assoc
thf(fact_7087_and_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ C )
      = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).

% and.assoc
thf(fact_7088_and_Ocommute,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [A4: int,B2: int] : ( bit_se725231765392027082nd_int @ B2 @ A4 ) ) ) ).

% and.commute
thf(fact_7089_and_Ocommute,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [A4: nat,B2: nat] : ( bit_se727722235901077358nd_nat @ B2 @ A4 ) ) ) ).

% and.commute
thf(fact_7090_and_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( bit_se725231765392027082nd_int @ B @ ( bit_se725231765392027082nd_int @ A @ C ) )
      = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).

% and.left_commute
thf(fact_7091_and_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( bit_se727722235901077358nd_nat @ B @ ( bit_se727722235901077358nd_nat @ A @ C ) )
      = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).

% and.left_commute
thf(fact_7092_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ A @ B )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
        & ( B
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_7093_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( bit_se725231765392027082nd_int @ A @ B )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ( A
          = ( uminus_uminus_int @ one_one_int ) )
        & ( B
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_7094_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_7095_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_7096_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_7097_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_7098_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_7099_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_7100_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_7101_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X2: real,Y5: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y5 ) ) ) ) ).

% minus_real_def
thf(fact_7102_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).

% and_less_eq
thf(fact_7103_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_7104_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_7105_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_7106_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_7107_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_7108_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_7109_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_7110_even__and__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_7111_even__and__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_7112_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).

% even_and_iff_int
thf(fact_7113_and__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_7114_and__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_7115_and__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_7116_one__and__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_7117_one__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_7118_one__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_7119_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_7120_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_7121_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_7122_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X2: real] : ( minus_minus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_7123_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_7124_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_7125_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_7126_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_7127_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L3: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_7128_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L3: int] :
          ( if_int
          @ ( ( K3 = zero_zero_int )
            | ( L3 = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            @ L3
            @ ( if_int
              @ ( L3
                = ( 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 @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_7129_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_7130_VEBTi_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( size_size_VEBT_VEBTi @ ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_a6397454172108246045_VEBTi @ size_size_VEBT_VEBTi @ X13 ) @ ( size_size_VEBT_VEBTi @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBTi.size(3)
thf(fact_7131_ln__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X )
          = ( suminf_real
            @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N4 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_7132_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_7133_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_7134_obtain__set__pred,axiom,
    ! [Z: nat,X: nat,A2: set_nat] :
      ( ( ord_less_nat @ Z @ X )
     => ( ( vEBT_VEBT_min_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ A2 )
         => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A2 @ X @ X_1 ) ) ) ) ).

% obtain_set_pred
thf(fact_7135_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_7136_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ Xs )
              & ( ord_less_nat @ A @ X4 ) ) ) ) ).

% succ_none_empty
thf(fact_7137_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ Xs )
              & ( ord_less_nat @ X4 @ A ) ) ) ) ).

% pred_none_empty
thf(fact_7138_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_7139_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_7140_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_7141_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_7142_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A2: set_nat,B3: set_nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ B3 )
         => ( ( A2 = B3 )
           => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A2 @ X @ X_1 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_7143_abs__0,axiom,
    ( ( abs_abs_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% abs_0
thf(fact_7144_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_7145_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_7146_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_7147_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_7148_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_7149_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_7150_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_7151_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_7152_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_7153_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_7154_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_7155_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_7156_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_7157_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_7158_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_7159_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_7160_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_numeral
thf(fact_7161_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_numeral
thf(fact_7162_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_numeral
thf(fact_7163_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_7164_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_7165_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_7166_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_7167_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_7168_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_7169_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_7170_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_7171_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_7172_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).

% List.finite_set
thf(fact_7173_List_Ofinite__set,axiom,
    ! [Xs: list_real] : ( finite_finite_real @ ( set_real2 @ Xs ) ) ).

% List.finite_set
thf(fact_7174_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_7175_List_Ofinite__set,axiom,
    ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).

% List.finite_set
thf(fact_7176_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).

% List.finite_set
thf(fact_7177_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_7178_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_7179_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_7180_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_7181_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_7182_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_7183_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_7184_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_7185_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_7186_abs__minus,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( abs_abs_complex @ A ) ) ).

% abs_minus
thf(fact_7187_abs__minus,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus
thf(fact_7188_abs__minus,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus
thf(fact_7189_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
      = ( dvd_dvd_real @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_7190_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
      = ( dvd_dvd_int @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_7191_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
      = ( dvd_dvd_real @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_7192_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
      = ( dvd_dvd_int @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_7193_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% abs_of_nat
thf(fact_7194_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_7195_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
      = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).

% of_int_abs
thf(fact_7196_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
      = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).

% of_int_abs
thf(fact_7197_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_rat @ ( abs_abs_int @ X ) )
      = ( abs_abs_rat @ ( ring_1_of_int_rat @ X ) ) ) ).

% of_int_abs
thf(fact_7198_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% abs_bool_eq
thf(fact_7199_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% abs_bool_eq
thf(fact_7200_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% abs_bool_eq
thf(fact_7201_tanh__0,axiom,
    ( ( tanh_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% tanh_0
thf(fact_7202_tanh__0,axiom,
    ( ( tanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% tanh_0
thf(fact_7203_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).

% finite_atLeastLessThan
thf(fact_7204_tanh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( ( tanh_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% tanh_real_zero_iff
thf(fact_7205_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% tanh_real_less_iff
thf(fact_7206_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_7207_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_7208_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_7209_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_7210_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_7211_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_7212_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_7213_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_7214_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_7215_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_7216_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_7217_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_7218_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_7219_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_7220_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_7221_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_7222_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_7223_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_7224_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_neg_numeral
thf(fact_7225_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_7226_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_7227_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_7228_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_7229_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_7230_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_7231_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_7232_infinite__Ico__iff,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A @ B ) ) )
      = ( ord_less_real @ A @ B ) ) ).

% infinite_Ico_iff
thf(fact_7233_infinite__Ico__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Ico_iff
thf(fact_7234_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_7235_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_7236_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_7237_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_7238_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% tanh_real_pos_iff
thf(fact_7239_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% tanh_real_neg_iff
thf(fact_7240_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% tanh_real_nonneg_iff
thf(fact_7241_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_7242_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_7243_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_7244_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_7245_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_7246_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_7247_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_7248_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_7249_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_7250_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_7251_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_7252_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_7253_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_7254_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_7255_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_7256_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_7257_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_7258_power2__abs,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_7259_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_7260_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_7261_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_7262_abs__power2,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_7263_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_7264_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_7265_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7266_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7267_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_7268_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_7269_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_7270_power__even__abs__numeral,axiom,
    ! [W: num,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_7271_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_7272_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_7273_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_7274_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_7275_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_7276_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_7277_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_7278_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_7279_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_7280_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_7281_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_7282_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_7283_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_7284_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_7285_power__abs,axiom,
    ! [A: rat,N: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
      = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).

% power_abs
thf(fact_7286_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( ( abs_abs_real @ L )
        = ( abs_abs_real @ K ) )
     => ( dvd_dvd_real @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_7287_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( ( abs_abs_int @ L )
        = ( abs_abs_int @ K ) )
     => ( dvd_dvd_int @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_7288_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( abs_abs_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
thf(fact_7289_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_7290_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_7291_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_7292_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_7293_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_7294_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_7295_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_7296_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_7297_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_7298_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_7299_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_7300_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_7301_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_7302_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_7303_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N8: set_nat] :
        ? [M5: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ N8 )
         => ( ord_less_nat @ X2 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_7304_bounded__nat__set__is__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ N5 )
         => ( ord_less_nat @ X3 @ N ) )
     => ( finite_finite_nat @ N5 ) ) ).

% bounded_nat_set_is_finite
thf(fact_7305_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_7306_finite__list,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ? [Xs3: list_real] :
          ( ( set_real2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_7307_finite__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_7308_finite__list,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_7309_finite__list,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_7310_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_7311_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7312_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
              & ( ( size_s3451745648224563538omplex @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7313_finite__lists__length__eq,axiom,
    ! [A2: set_real,N: nat] :
      ( ( finite_finite_real @ A2 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs2: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_real @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7314_finite__lists__length__eq,axiom,
    ! [A2: set_o,N: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs2: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_o @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7315_finite__lists__length__eq,axiom,
    ! [A2: set_int,N: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_int @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7316_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_nat @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7317_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_7318_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_7319_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_7320_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_7321_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_7322_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_7323_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_7324_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_7325_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7326_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7327_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7328_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7329_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_7330_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_7331_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_7332_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
     => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D2 )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_7333_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D2 )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_7334_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D2 )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_7335_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_7336_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_7337_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_7338_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_7339_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_7340_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_7341_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_7342_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_7343_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_7344_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_7345_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_7346_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_7347_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_7348_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_7349_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_7350_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_7351_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_7352_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_7353_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_7354_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_7355_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_7356_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_7357_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_7358_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_7359_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_7360_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_7361_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_7362_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_7363_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_7364_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_7365_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_7366_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_7367_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_real_def
thf(fact_7368_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_7369_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_7370_infinite__Ico,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A @ B ) ) ) ).

% infinite_Ico
thf(fact_7371_infinite__Ico,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) ) ).

% infinite_Ico
thf(fact_7372_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7373_finite__lists__length__le,axiom,
    ! [A2: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7374_finite__lists__length__le,axiom,
    ! [A2: set_real,N: nat] :
      ( ( finite_finite_real @ A2 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs2: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7375_finite__lists__length__le,axiom,
    ! [A2: set_o,N: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs2: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7376_finite__lists__length__le,axiom,
    ! [A2: set_int,N: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7377_finite__lists__length__le,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7378_tanh__real__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_7379_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_7380_filter__preserves__multiset,axiom,
    ! [M8: product_prod_int_int > nat,P: product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M8 @ X2 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_7381_filter__preserves__multiset,axiom,
    ! [M8: nat > nat,P: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M8 @ X2 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_7382_filter__preserves__multiset,axiom,
    ! [M8: int > nat,P: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M8 @ X2 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_7383_filter__preserves__multiset,axiom,
    ! [M8: complex > nat,P: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M8 @ X2 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_7384_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_7385_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_7386_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_7387_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_7388_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_7389_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_7390_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_7391_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_7392_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_7393_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_7394_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_7395_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_7396_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_7397_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_7398_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_7399_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_7400_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_7401_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_7402_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_7403_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_7404_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_7405_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_7406_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_7407_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_7408_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_7409_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_7410_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_7411_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_7412_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_7413_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_7414_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_7415_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_7416_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_7417_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_7418_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_7419_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_7420_abs__of__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_neg
thf(fact_7421_abs__of__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_neg
thf(fact_7422_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_7423_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_7424_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D2 ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7425_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D2 ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7426_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D2 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7427_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_7428_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_7429_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_7430_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_7431_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_7432_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_7433_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_7434_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_7435_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_7436_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_7437_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 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D3 )
               => ( ( ord_less_real @ A @ Y4 )
                  & ( ord_less_real @ Y4 @ B ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_7438_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_7439_add__mset__in__multiset,axiom,
    ! [M8: product_prod_int_int > nat,A: product_prod_int_int] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X2 = A ) @ ( suc @ ( M8 @ X2 ) ) @ ( M8 @ X2 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7440_add__mset__in__multiset,axiom,
    ! [M8: nat > nat,A: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X2 = A ) @ ( suc @ ( M8 @ X2 ) ) @ ( M8 @ X2 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7441_add__mset__in__multiset,axiom,
    ! [M8: int > nat,A: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X2 = A ) @ ( suc @ ( M8 @ X2 ) ) @ ( M8 @ X2 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7442_add__mset__in__multiset,axiom,
    ! [M8: complex > nat,A: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X2 = A ) @ ( suc @ ( M8 @ X2 ) ) @ ( M8 @ X2 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_7443_diff__preserves__multiset,axiom,
    ! [M8: product_prod_int_int > nat,N5: product_prod_int_int > nat] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X2 ) @ ( N5 @ X2 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_7444_diff__preserves__multiset,axiom,
    ! [M8: nat > nat,N5: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X2 ) @ ( N5 @ X2 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_7445_diff__preserves__multiset,axiom,
    ! [M8: int > nat,N5: int > nat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X2 ) @ ( N5 @ X2 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_7446_diff__preserves__multiset,axiom,
    ! [M8: complex > nat,N5: complex > nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X2 ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X2 ) @ ( N5 @ X2 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_7447_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D: nat] : ( dvd_dvd_nat @ D @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_7448_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N5 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_7449_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N5 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_7450_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_7451_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_7452_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_7453_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_7454_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_7455_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_7456_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_7457_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_7458_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_7459_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_7460_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_7461_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 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D3 )
               => ( ( ord_less_eq_real @ A @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_7462_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z7: real] :
              ( ( power_power_real @ Z7 @ N )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_7463_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z7: complex] :
              ( ( power_power_complex @ Z7 @ N )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_7464_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_7465_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_7466_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_7467_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_7468_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_7469_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_7470_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_7471_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_7472_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_7473_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_7474_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_7475_power__even__abs,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
        = ( power_power_rat @ A @ N ) ) ) ).

% power_even_abs
thf(fact_7476_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_7477_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_7478_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_7479_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_7480_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_7481_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_7482_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_7483_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_7484_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_7485_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_7486_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_7487_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_7488_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_7489_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_7490_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_7491_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_7492_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N4: nat] :
          ( if_nat
          @ ( ( M5 = zero_zero_nat )
            | ( N4 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_7493_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_7494_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_7495_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_7496_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_7497_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_7498_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_7499_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_7500_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_7501_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_7502_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X: code_integer] :
      ( ! [X3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
         => ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7503_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
         => ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7504_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
         => ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7505_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X: int] :
      ( ! [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
         => ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7506_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_7507_suminf__zero,axiom,
    ( ( suminf_complex
      @ ^ [N4: nat] : zero_zero_complex )
    = zero_zero_complex ) ).

% suminf_zero
thf(fact_7508_suminf__zero,axiom,
    ( ( suminf_real
      @ ^ [N4: nat] : zero_zero_real )
    = zero_zero_real ) ).

% suminf_zero
thf(fact_7509_suminf__zero,axiom,
    ( ( suminf_nat
      @ ^ [N4: nat] : zero_zero_nat )
    = zero_zero_nat ) ).

% suminf_zero
thf(fact_7510_suminf__zero,axiom,
    ( ( suminf_int
      @ ^ [N4: nat] : zero_zero_int )
    = zero_zero_int ) ).

% suminf_zero
thf(fact_7511_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( vEBT_size_VEBTi @ ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_a6397454172108246045_VEBTi @ vEBT_size_VEBTi @ X13 ) @ ( vEBT_size_VEBTi @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBTi.size_gen(1)
thf(fact_7512_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_eq_int @ A @ I2 )
            & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_7513_finite__interval__int4,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_int @ A @ I2 )
            & ( ord_less_int @ I2 @ B ) ) ) ) ).

% finite_interval_int4
thf(fact_7514_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z7: complex] :
              ( ( power_power_complex @ Z7 @ N )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_7515_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_eq_int @ A @ I2 )
            & ( ord_less_int @ I2 @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_7516_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_int @ A @ I2 )
            & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_7517_zdvd1__eq,axiom,
    ! [X: int] :
      ( ( dvd_dvd_int @ X @ one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_7518_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_7519_zdvd__antisym__abs,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( abs_abs_int @ A )
          = ( abs_abs_int @ B ) ) ) ) ).

% zdvd_antisym_abs
thf(fact_7520_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
        = one_one_int )
     => ( ( abs_abs_int @ M )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_7521_finite__maxlen,axiom,
    ! [M8: set_list_real] :
      ( ( finite306553202115118035t_real @ M8 )
     => ? [N2: nat] :
        ! [X4: list_real] :
          ( ( member_list_real @ X4 @ M8 )
         => ( ord_less_nat @ ( size_size_list_real @ X4 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_7522_finite__maxlen,axiom,
    ! [M8: set_list_o] :
      ( ( finite_finite_list_o @ M8 )
     => ? [N2: nat] :
        ! [X4: list_o] :
          ( ( member_list_o @ X4 @ M8 )
         => ( ord_less_nat @ ( size_size_list_o @ X4 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_7523_finite__maxlen,axiom,
    ! [M8: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M8 )
     => ? [N2: nat] :
        ! [X4: list_nat] :
          ( ( member_list_nat @ X4 @ M8 )
         => ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_7524_finite__maxlen,axiom,
    ! [M8: set_list_int] :
      ( ( finite3922522038869484883st_int @ M8 )
     => ? [N2: nat] :
        ! [X4: list_int] :
          ( ( member_list_int @ X4 @ M8 )
         => ( ord_less_nat @ ( size_size_list_int @ X4 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_7525_abs__div,axiom,
    ! [Y: int,X: int] :
      ( ( dvd_dvd_int @ Y @ X )
     => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).

% abs_div
thf(fact_7526_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_7527_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [D: int] : ( dvd_dvd_int @ D @ I ) ) ) ) ).

% finite_divisors_int
thf(fact_7528_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I2: int] : ( if_int @ ( ord_less_int @ I2 @ zero_zero_int ) @ ( uminus_uminus_int @ I2 ) @ I2 ) ) ) ).

% zabs_def
thf(fact_7529_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).

% abs_mod_less
thf(fact_7530_dvd__imp__le__int,axiom,
    ! [I: int,D2: int] :
      ( ( I != zero_zero_int )
     => ( ( dvd_dvd_int @ D2 @ I )
       => ( ord_less_eq_int @ ( abs_abs_int @ D2 ) @ ( abs_abs_int @ I ) ) ) ) ).

% dvd_imp_le_int
thf(fact_7531_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
        = ( ( abs_abs_int @ N )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_7532_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_abs_add_iff
thf(fact_7533_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_add_abs_iff
thf(fact_7534_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ( ord_less_eq_nat @ M @ I3 )
            & ( ord_less_nat @ I3 @ N ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
           => ? [I3: nat] :
                ( ( ord_less_eq_nat @ M @ I3 )
                & ( ord_less_eq_nat @ I3 @ N )
                & ( ( F @ I3 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_7535_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) @ Z ) ) ).

% decr_lemma
thf(fact_7536_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( 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 ) @ D2 ) ) ) ) ).

% incr_lemma
thf(fact_7537_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( vEBT_size_VEBTi @ ( vEBT_Leafi @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBTi.size_gen(2)
thf(fact_7538_nat__ivt__aux,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_7539_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_7540_nat0__intermed__int__val,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_7541_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_7542_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_7543_set__encode__insert,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ N @ A2 )
       => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).

% set_encode_insert
thf(fact_7544_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_7545_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B5: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ! [X4: real] :
                    ( ( member_real @ X4 @ A8 )
                   => ( ord_less_real @ X4 @ B5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_real @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7546_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B5: rat,A8: set_rat] :
              ( ( finite_finite_rat @ A8 )
             => ( ! [X4: rat] :
                    ( ( member_rat @ X4 @ A8 )
                   => ( ord_less_rat @ X4 @ B5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_rat @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7547_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B5: num,A8: set_num] :
              ( ( finite_finite_num @ A8 )
             => ( ! [X4: num] :
                    ( ( member_num @ X4 @ A8 )
                   => ( ord_less_num @ X4 @ B5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_num @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7548_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ A8 )
                   => ( ord_less_nat @ X4 @ B5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_nat @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7549_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B5: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ! [X4: int] :
                    ( ( member_int @ X4 @ A8 )
                   => ( ord_less_int @ X4 @ B5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_int @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7550_finite__linorder__max__induct,axiom,
    ! [A2: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B5: code_integer,A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ! [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ A8 )
                   => ( ord_le6747313008572928689nteger @ X4 @ B5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_Code_integer @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_7551_arctan__zero__zero,axiom,
    ( ( arctan @ zero_zero_real )
    = zero_zero_real ) ).

% arctan_zero_zero
thf(fact_7552_arctan__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( ( arctan @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% arctan_eq_zero_iff
thf(fact_7553_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% zero_less_arctan_iff
thf(fact_7554_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% arctan_less_zero_iff
thf(fact_7555_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_7556_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_arctan_iff
thf(fact_7557_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_7558_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% arctan_less_iff
thf(fact_7559_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone
thf(fact_7560_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
thf(fact_7561_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_7562_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_7563_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_7564_arctan__ubound,axiom,
    ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_7565_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_7566_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_7567_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_7568_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( nat_set_encode @ A2 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_7569_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_7570_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_7571_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_7572_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_7573_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_7574_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_7575_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_7576_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_7577_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_7578_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_7579_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_7580_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_7581_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_7582_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_7583_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).

% even_set_encode_iff
thf(fact_7584_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ S2 )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S2 )
                  & ( ord_less_real @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_7585_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( ( finite_finite_rat @ S2 )
     => ( ( S2 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ S2 )
            & ~ ? [Xa: rat] :
                  ( ( member_rat @ Xa @ S2 )
                  & ( ord_less_rat @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_7586_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( ( finite_finite_num @ S2 )
     => ( ( S2 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ S2 )
            & ~ ? [Xa: num] :
                  ( ( member_num @ Xa @ S2 )
                  & ( ord_less_num @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_7587_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ S2 )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S2 )
                  & ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_7588_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( ( finite_finite_int @ S2 )
     => ( ( S2 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ S2 )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S2 )
                  & ( ord_less_int @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_7589_ex__min__if__finite,axiom,
    ! [S2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( S2 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ S2 )
            & ~ ? [Xa: code_integer] :
                  ( ( member_Code_integer @ Xa @ S2 )
                  & ( ord_le6747313008572928689nteger @ Xa @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_7590_infinite__growing,axiom,
    ! [X9: set_real] :
      ( ( X9 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ X9 )
           => ? [Xa: real] :
                ( ( member_real @ Xa @ X9 )
                & ( ord_less_real @ X3 @ Xa ) ) )
       => ~ ( finite_finite_real @ X9 ) ) ) ).

% infinite_growing
thf(fact_7591_infinite__growing,axiom,
    ! [X9: set_rat] :
      ( ( X9 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( ( member_rat @ X3 @ X9 )
           => ? [Xa: rat] :
                ( ( member_rat @ Xa @ X9 )
                & ( ord_less_rat @ X3 @ Xa ) ) )
       => ~ ( finite_finite_rat @ X9 ) ) ) ).

% infinite_growing
thf(fact_7592_infinite__growing,axiom,
    ! [X9: set_num] :
      ( ( X9 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( ( member_num @ X3 @ X9 )
           => ? [Xa: num] :
                ( ( member_num @ Xa @ X9 )
                & ( ord_less_num @ X3 @ Xa ) ) )
       => ~ ( finite_finite_num @ X9 ) ) ) ).

% infinite_growing
thf(fact_7593_infinite__growing,axiom,
    ! [X9: set_nat] :
      ( ( X9 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X9 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X9 )
                & ( ord_less_nat @ X3 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X9 ) ) ) ).

% infinite_growing
thf(fact_7594_infinite__growing,axiom,
    ! [X9: set_int] :
      ( ( X9 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ X9 )
           => ? [Xa: int] :
                ( ( member_int @ Xa @ X9 )
                & ( ord_less_int @ X3 @ Xa ) ) )
       => ~ ( finite_finite_int @ X9 ) ) ) ).

% infinite_growing
thf(fact_7595_infinite__growing,axiom,
    ! [X9: set_Code_integer] :
      ( ( X9 != bot_bo3990330152332043303nteger )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ X9 )
           => ? [Xa: code_integer] :
                ( ( member_Code_integer @ Xa @ X9 )
                & ( ord_le6747313008572928689nteger @ X3 @ Xa ) ) )
       => ~ ( finite6017078050557962740nteger @ X9 ) ) ) ).

% infinite_growing
thf(fact_7596_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_7597_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B5: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ! [X4: real] :
                    ( ( member_real @ X4 @ A8 )
                   => ( ord_less_real @ B5 @ X4 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_real @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7598_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B5: rat,A8: set_rat] :
              ( ( finite_finite_rat @ A8 )
             => ( ! [X4: rat] :
                    ( ( member_rat @ X4 @ A8 )
                   => ( ord_less_rat @ B5 @ X4 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_rat @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7599_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B5: num,A8: set_num] :
              ( ( finite_finite_num @ A8 )
             => ( ! [X4: num] :
                    ( ( member_num @ X4 @ A8 )
                   => ( ord_less_num @ B5 @ X4 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_num @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7600_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ A8 )
                   => ( ord_less_nat @ B5 @ X4 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_nat @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7601_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B5: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ! [X4: int] :
                    ( ( member_int @ X4 @ A8 )
                   => ( ord_less_int @ B5 @ X4 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_int @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7602_finite__linorder__min__induct,axiom,
    ! [A2: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B5: code_integer,A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ! [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ A8 )
                   => ( ord_le6747313008572928689nteger @ B5 @ X4 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_Code_integer @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_7603_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_7604_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_7605_cos__pi__eq__zero,axiom,
    ! [M: nat] :
      ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = zero_zero_real ) ).

% cos_pi_eq_zero
thf(fact_7606_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ( ~ ( finite_finite_int @ S2 ) )
      = ( ! [M5: int] :
          ? [N4: int] :
            ( ( ord_less_int @ M5 @ ( abs_abs_int @ N4 ) )
            & ( member_int @ N4 @ S2 ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_7607_vebt__buildupi__refines,axiom,
    ! [N: nat] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_V739175172307565963ildupi @ N ) ) ).

% vebt_buildupi_refines
thf(fact_7608_refines__replicate,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,F5: heap_T8145700208782473153_VEBTi,N: nat] :
      ( ( refine5565527176597971370_VEBTi @ F @ F5 )
     => ( refine3700189196150522554_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N @ F ) @ ( vEBT_V1859673955506687831_VEBTi @ N @ F5 ) ) ) ).

% refines_replicate
thf(fact_7609_sin__zero,axiom,
    ( ( sin_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sin_zero
thf(fact_7610_sin__zero,axiom,
    ( ( sin_real @ zero_zero_real )
    = zero_zero_real ) ).

% sin_zero
thf(fact_7611_summable__zero,axiom,
    ( summable_complex
    @ ^ [N4: nat] : zero_zero_complex ) ).

% summable_zero
thf(fact_7612_summable__zero,axiom,
    ( summable_real
    @ ^ [N4: nat] : zero_zero_real ) ).

% summable_zero
thf(fact_7613_summable__zero,axiom,
    ( summable_nat
    @ ^ [N4: nat] : zero_zero_nat ) ).

% summable_zero
thf(fact_7614_summable__zero,axiom,
    ( summable_int
    @ ^ [N4: nat] : zero_zero_int ) ).

% summable_zero
thf(fact_7615_summable__single,axiom,
    ! [I: nat,F: nat > complex] :
      ( summable_complex
      @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex ) ) ).

% summable_single
thf(fact_7616_summable__single,axiom,
    ! [I: nat,F: nat > real] :
      ( summable_real
      @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).

% summable_single
thf(fact_7617_summable__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( summable_nat
      @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).

% summable_single
thf(fact_7618_summable__single,axiom,
    ! [I: nat,F: nat > int] :
      ( summable_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).

% summable_single
thf(fact_7619_summable__iff__shift,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real
        @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K ) ) )
      = ( summable_real @ F ) ) ).

% summable_iff_shift
thf(fact_7620_cos__zero,axiom,
    ( ( cos_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cos_zero
thf(fact_7621_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_7622_sin__pi,axiom,
    ( ( sin_real @ pi )
    = zero_zero_real ) ).

% sin_pi
thf(fact_7623_summable__cmult__iff,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ C @ ( F @ N4 ) ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7624_summable__cmult__iff,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7625_summable__divide__iff,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ C ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_divide_iff
thf(fact_7626_summable__divide__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_divide_iff
thf(fact_7627_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_complex
        @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite
thf(fact_7628_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite
thf(fact_7629_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite
thf(fact_7630_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite
thf(fact_7631_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_complex
        @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite_set
thf(fact_7632_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite_set
thf(fact_7633_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite_set
thf(fact_7634_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite_set
thf(fact_7635_cos__periodic__pi2,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_periodic_pi2
thf(fact_7636_cos__periodic__pi,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_periodic_pi
thf(fact_7637_sin__periodic__pi2,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_periodic_pi2
thf(fact_7638_sin__periodic__pi,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_periodic_pi
thf(fact_7639_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_7640_sin__npi2,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_7641_sin__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_7642_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_7643_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_7644_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_7645_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_7646_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_7647_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_7648_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_7649_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_7650_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_7651_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_7652_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_7653_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_7654_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_7655_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_7656_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_7657_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_7658_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_7659_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_7660_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_7661_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_7662_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_7663_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_7664_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_7665_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_7666_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_T8145700208782473153_VEBTi,F12: $o > $o > heap_T8145700208782473153_VEBTi,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F23: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi] :
      ( ( Ti = Ti2 )
     => ( ! [A6: $o,B5: $o] : ( refine5565527176597971370_VEBTi @ ( F1 @ A6 @ B5 ) @ ( F12 @ A6 @ B5 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine5565527176597971370_VEBTi @ ( F22 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( F23 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
         => ( refine5565527176597971370_VEBTi @ ( vEBT_c6028912655521741485_VEBTi @ F22 @ F1 @ Ti ) @ ( vEBT_c6028912655521741485_VEBTi @ F23 @ F12 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_7667_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_7668_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_7669_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_7670_polar__Ex,axiom,
    ! [X: real,Y: real] :
    ? [R: real,A6: real] :
      ( ( X
        = ( times_times_real @ R @ ( cos_real @ A6 ) ) )
      & ( Y
        = ( times_times_real @ R @ ( sin_real @ A6 ) ) ) ) ).

% polar_Ex
thf(fact_7671_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_7672_summable__const__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex
        @ ^ [Uu3: nat] : C )
      = ( C = zero_zero_complex ) ) ).

% summable_const_iff
thf(fact_7673_summable__const__iff,axiom,
    ! [C: real] :
      ( ( summable_real
        @ ^ [Uu3: nat] : C )
      = ( C = zero_zero_real ) ) ).

% summable_const_iff
thf(fact_7674_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_7675_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_7676_summable__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N4: nat] : ( plus_plus_real @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).

% summable_add
thf(fact_7677_summable__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( summable_nat
          @ ^ [N4: nat] : ( plus_plus_nat @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).

% summable_add
thf(fact_7678_summable__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( summable_int
          @ ^ [N4: nat] : ( plus_plus_int @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).

% summable_add
thf(fact_7679_summable__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ C ) ) ) ).

% summable_mult2
thf(fact_7680_summable__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) ) ) ) ).

% summable_mult
thf(fact_7681_summable__Suc__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
      = ( summable_real @ F ) ) ).

% summable_Suc_iff
thf(fact_7682_summable__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N4: nat] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ C ) ) ) ).

% summable_divide
thf(fact_7683_summable__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ C ) ) ) ).

% summable_divide
thf(fact_7684_summable__ignore__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_7685_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_7686_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_7687_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_7688_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_7689_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_7690_powser__insidea,axiom,
    ! [F: nat > real,X: real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ X @ N4 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
       => ( summable_real
          @ ^ [N4: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7691_powser__insidea,axiom,
    ! [F: nat > complex,X: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ X @ N4 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
       => ( summable_real
          @ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7692_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
      & ( ord_less_eq_real @ Y3 @ pi )
      & ( ( sin_real @ Y3 )
        = ( sin_real @ X ) )
      & ( ( cos_real @ Y3 )
        = ( cos_real @ X ) ) ) ).

% sincos_principal_value
thf(fact_7693_summable__finite,axiom,
    ! [N5: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_complex ) )
       => ( summable_complex @ F ) ) ) ).

% summable_finite
thf(fact_7694_summable__finite,axiom,
    ! [N5: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_real ) )
       => ( summable_real @ F ) ) ) ).

% summable_finite
thf(fact_7695_summable__finite,axiom,
    ! [N5: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_nat ) )
       => ( summable_nat @ F ) ) ) ).

% summable_finite
thf(fact_7696_summable__finite,axiom,
    ! [N5: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_int ) )
       => ( summable_int @ F ) ) ) ).

% summable_finite
thf(fact_7697_sin__x__le__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).

% sin_x_le_x
thf(fact_7698_summable__mult__D,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ C @ ( F @ N4 ) ) )
     => ( ( C != zero_zero_complex )
       => ( summable_complex @ F ) ) ) ).

% summable_mult_D
thf(fact_7699_summable__mult__D,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) ) )
     => ( ( C != zero_zero_real )
       => ( summable_real @ F ) ) ) ).

% summable_mult_D
thf(fact_7700_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_7701_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_7702_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_7703_cos__arctan__not__zero,axiom,
    ! [X: real] :
      ( ( cos_real @ ( arctan @ X ) )
     != zero_zero_real ) ).

% cos_arctan_not_zero
thf(fact_7704_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
            @ ^ [N4: nat] : ( plus_plus_real @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7705_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
            @ ^ [N4: nat] : ( plus_plus_nat @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7706_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
            @ ^ [N4: nat] : ( plus_plus_int @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7707_suminf__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( times_times_real @ ( suminf_real @ F ) @ C )
        = ( suminf_real
          @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ C ) ) ) ) ).

% suminf_mult2
thf(fact_7708_suminf__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) ) )
        = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).

% suminf_mult
thf(fact_7709_suminf__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N4: nat] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ C ) )
        = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_7710_suminf__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ C ) )
        = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_7711_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_7712_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_7713_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_7714_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_7715_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_7716_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_7717_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_7718_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_7719_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 )
          = ( ! [N4: nat] :
                ( ( F @ N4 )
                = zero_zero_real ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7720_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 )
          = ( ! [N4: nat] :
                ( ( F @ N4 )
                = zero_zero_nat ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7721_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 )
          = ( ! [N4: nat] :
                ( ( F @ N4 )
                = zero_zero_int ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7722_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_7723_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_7724_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_7725_summable__zero__power_H,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) ) ).

% summable_zero_power'
thf(fact_7726_summable__zero__power_H,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) ) ).

% summable_zero_power'
thf(fact_7727_summable__zero__power_H,axiom,
    ! [F: nat > int] :
      ( summable_int
      @ ^ [N4: nat] : ( times_times_int @ ( F @ N4 ) @ ( power_power_int @ zero_zero_int @ N4 ) ) ) ).

% summable_zero_power'
thf(fact_7728_summable__0__powser,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) ) ).

% summable_0_powser
thf(fact_7729_summable__0__powser,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) ) ).

% summable_0_powser
thf(fact_7730_sin__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ pi )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero
thf(fact_7731_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).

% sin_x_ge_neg_x
thf(fact_7732_sin__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_ge_zero
thf(fact_7733_powser__split__head_I3_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
     => ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ).

% powser_split_head(3)
thf(fact_7734_powser__split__head_I3_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
     => ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) ) ) ).

% powser_split_head(3)
thf(fact_7735_summable__powser__split__head,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) )
      = ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ).

% summable_powser_split_head
thf(fact_7736_summable__powser__split__head,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) )
      = ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) ) ) ).

% summable_powser_split_head
thf(fact_7737_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ( cos_real @ X )
                = ( cos_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_7738_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_7739_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_7740_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > complex,M: nat,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N4 @ M ) ) @ ( power_power_complex @ Z @ N4 ) ) )
      = ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7741_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > real,M: nat,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N4 @ M ) ) @ ( power_power_real @ Z @ N4 ) ) )
      = ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7742_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_7743_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_7744_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_7745_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_7746_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_7747_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_7748_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_7749_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_7750_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_7751_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_7752_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_7753_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_7754_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_7755_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_7756_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_7757_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_7758_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 ) )
          = ( ? [I2: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7759_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 ) )
          = ( ? [I2: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7760_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 ) )
          = ( ? [I2: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I2 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7761_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_7762_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_7763_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_7764_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_7765_powser__inside,axiom,
    ! [F: nat > real,X: real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ X @ N4 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
       => ( summable_real
          @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) ) ) ) ).

% powser_inside
thf(fact_7766_powser__inside,axiom,
    ! [F: nat > complex,X: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ X @ N4 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
       => ( summable_complex
          @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ) ).

% powser_inside
thf(fact_7767_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_7768_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_7769_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
     => ( ( ord_less_real @ X @ pi )
       => ( ( ( sin_real @ X )
            = zero_zero_real )
         => ( X = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
thf(fact_7770_suminf__split__head,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_7771_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_7772_summable__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ C ) ) ) ).

% summable_geometric
thf(fact_7773_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_7774_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_7775_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
thf(fact_7776_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_7777_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I2: int] :
            ( X
            = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_7778_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 )
       => ? [T4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
            & ( ord_less_eq_real @ T4 @ pi )
            & ( X
              = ( cos_real @ T4 ) )
            & ( Y
              = ( sin_real @ T4 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_7779_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_7780_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_7781_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_7782_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_7783_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_7784_cos__is__zero,axiom,
    ? [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
      & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      & ( ( cos_real @ X3 )
        = zero_zero_real )
      & ! [Y4: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
            & ( ord_less_eq_real @ Y4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y4 )
              = zero_zero_real ) )
         => ( Y4 = X3 ) ) ) ).

% cos_is_zero
thf(fact_7785_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_7786_cos__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
            & ( ord_less_eq_real @ X3 @ pi )
            & ( ( cos_real @ X3 )
              = Y )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ pi )
                  & ( ( cos_real @ Y4 )
                    = Y ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% cos_total
thf(fact_7787_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 )
         => ? [T4: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T4 )
              & ( ord_less_eq_real @ T4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X
                = ( cos_real @ T4 ) )
              & ( Y
                = ( sin_real @ T4 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_7788_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 )
     => ? [T4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T4 )
          & ( ord_less_eq_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X
            = ( cos_real @ T4 ) )
          & ( Y
            = ( sin_real @ T4 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_7789_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 )
     => ~ ! [T4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
           => ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X
                  = ( cos_real @ T4 ) )
               => ( Y
                 != ( sin_real @ T4 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_7790_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
     => ( ( suminf_complex
          @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7791_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
     => ( ( suminf_real
          @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7792_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) )
          @ Z )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7793_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) )
          @ Z )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7794_suminf__exist__split,axiom,
    ! [R2: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_real @ F )
       => ? [N9: nat] :
          ! [N10: nat] :
            ( ( ord_less_eq_nat @ N9 @ N10 )
           => ( ord_less_real
              @ ( real_V7735802525324610683m_real
                @ ( suminf_real
                  @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N10 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_7795_suminf__exist__split,axiom,
    ! [R2: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_complex @ F )
       => ? [N9: nat] :
          ! [N10: nat] :
            ( ( ord_less_eq_nat @ N9 @ N10 )
           => ( ord_less_real
              @ ( real_V1022390504157884413omplex
                @ ( suminf_complex
                  @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N10 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_7796_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_7797_summable__power__series,axiom,
    ! [F: nat > real,Z: real] :
      ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
     => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( summable_real
              @ ^ [I2: nat] : ( times_times_real @ ( F @ I2 ) @ ( power_power_real @ Z @ I2 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_7798_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: nat > complex,M8: 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 ) ) @ M8 )
         => ( summable_real
            @ ^ [N4: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N4 ) ) @ ( power_power_real @ R2 @ N4 ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_7799_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_7800_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_7801_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_7802_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_7803_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_7804_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_7805_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_7806_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_7807_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_7808_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_7809_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_7810_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_7811_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_7812_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_7813_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_7814_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X2: int] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_7815_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_7816_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_7817_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_7818_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_7819_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M3: nat] :
          ( ( ord_less_nat @ K @ M3 )
         => ? [N10: nat] :
              ( ( ord_less_nat @ M3 @ N10 )
              & ( member_nat @ N10 @ S2 ) ) )
     => ~ ( finite_finite_nat @ S2 ) ) ).

% unbounded_k_infinite
thf(fact_7820_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M5: nat] :
          ? [N4: nat] :
            ( ( ord_less_nat @ M5 @ N4 )
            & ( member_nat @ N4 @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_7821_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_7822_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_7823_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_7824_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_7825_sin__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
            & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X3 )
              = Y )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y4 )
                    = Y ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% sin_total
thf(fact_7826_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_7827_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_7828_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X2: nat] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X2: nat] :
            ( X
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_7829_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_7830_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I2: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_7831_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [I2: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_7832_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_7833_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [N4: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N4: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_7834_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_7835_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [N4: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N4: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_7836_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_7837_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_7838_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
           => ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T4 ) @ ( sin_real @ T4 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_7839_and__int_Opelims,axiom,
    ! [X: int,Xa3: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa3 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa3 ) )
       => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa3 @ ( 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 ) ) @ Xa3 ) ) ) ) ) )
              & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa3 @ ( 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 ) ) @ Xa3 ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa3 ) ) ) ) ) ).

% and_int.pelims
thf(fact_7840_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
     => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_7841_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_7842_intind,axiom,
    ! [I: nat,N: nat,P: vEBT_VEBTi > $o,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X )
       => ( P @ ( nth_VEBT_VEBTi @ ( replicate_VEBT_VEBTi @ N @ X ) @ I ) ) ) ) ).

% intind
thf(fact_7843_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_7844_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_7845_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_7846_repli__cons__repl,axiom,
    ! [Q: assn,X: heap_Time_Heap_o,A2: vEBT_VEBT > $o > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_hoare_triple_o @ Q @ X
        @ ^ [R5: $o] : ( times_times_assn @ Q @ ( A2 @ Y @ R5 ) ) )
     => ( hoare_9089481587091695345list_o @ Q @ ( vEBT_V2326993469660664182atei_o @ N @ X )
        @ ^ [R5: list_o] : ( times_times_assn @ Q @ ( vEBT_L7489408758114837031VEBT_o @ A2 @ ( replicate_VEBT_VEBT @ N @ Y ) @ R5 ) ) ) ) ).

% repli_cons_repl
thf(fact_7847_repli__cons__repl,axiom,
    ! [Q: assn,X: heap_T2636463487746394924on_nat,A2: vEBT_VEBT > option_nat > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_7629718768684598413on_nat @ Q @ X
        @ ^ [R5: option_nat] : ( times_times_assn @ Q @ ( A2 @ Y @ R5 ) ) )
     => ( hoare_6480275734082232733on_nat @ Q @ ( vEBT_V792416675989592002on_nat @ N @ X )
        @ ^ [R5: list_option_nat] : ( times_times_assn @ Q @ ( vEBT_L8010285020845282001on_nat @ A2 @ ( replicate_VEBT_VEBT @ N @ Y ) @ R5 ) ) ) ) ).

% repli_cons_repl
thf(fact_7848_repli__cons__repl,axiom,
    ! [Q: assn,X: heap_Time_Heap_nat,A2: vEBT_VEBT > nat > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_3067605981109127869le_nat @ Q @ X
        @ ^ [R5: nat] : ( times_times_assn @ Q @ ( A2 @ Y @ R5 ) ) )
     => ( hoare_7964568885773372237st_nat @ Q @ ( vEBT_V7726092123322077554ei_nat @ N @ X )
        @ ^ [R5: list_nat] : ( times_times_assn @ Q @ ( vEBT_L8296926524756676353BT_nat @ A2 @ ( replicate_VEBT_VEBT @ N @ Y ) @ R5 ) ) ) ) ).

% repli_cons_repl
thf(fact_7849_repli__cons__repl,axiom,
    ! [Q: assn,X: heap_T8145700208782473153_VEBTi,A2: vEBT_VEBT > vEBT_VEBTi > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ Q @ X
        @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ Q @ ( A2 @ Y @ R5 ) ) )
     => ( hoare_3904069481286416050_VEBTi @ Q @ ( vEBT_V1859673955506687831_VEBTi @ N @ X )
        @ ^ [R5: list_VEBT_VEBTi] : ( times_times_assn @ Q @ ( vEBT_L6296928887356842470_VEBTi @ A2 @ ( replicate_VEBT_VEBT @ N @ Y ) @ R5 ) ) ) ) ).

% repli_cons_repl
thf(fact_7850_repli__emp,axiom,
    ! [X: heap_Time_Heap_o,A2: vEBT_VEBT > $o > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_hoare_triple_o @ one_one_assn @ X @ ( A2 @ Y ) )
     => ( hoare_9089481587091695345list_o @ one_one_assn @ ( vEBT_V2326993469660664182atei_o @ N @ X ) @ ( vEBT_L7489408758114837031VEBT_o @ A2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) ) ) ).

% repli_emp
thf(fact_7851_repli__emp,axiom,
    ! [X: heap_T2636463487746394924on_nat,A2: vEBT_VEBT > option_nat > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_7629718768684598413on_nat @ one_one_assn @ X @ ( A2 @ Y ) )
     => ( hoare_6480275734082232733on_nat @ one_one_assn @ ( vEBT_V792416675989592002on_nat @ N @ X ) @ ( vEBT_L8010285020845282001on_nat @ A2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) ) ) ).

% repli_emp
thf(fact_7852_repli__emp,axiom,
    ! [X: heap_Time_Heap_nat,A2: vEBT_VEBT > nat > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_3067605981109127869le_nat @ one_one_assn @ X @ ( A2 @ Y ) )
     => ( hoare_7964568885773372237st_nat @ one_one_assn @ ( vEBT_V7726092123322077554ei_nat @ N @ X ) @ ( vEBT_L8296926524756676353BT_nat @ A2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) ) ) ).

% repli_emp
thf(fact_7853_repli__emp,axiom,
    ! [X: heap_T8145700208782473153_VEBTi,A2: vEBT_VEBT > vEBT_VEBTi > assn,Y: vEBT_VEBT,N: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ one_one_assn @ X @ ( A2 @ Y ) )
     => ( hoare_3904069481286416050_VEBTi @ one_one_assn @ ( vEBT_V1859673955506687831_VEBTi @ N @ X ) @ ( vEBT_L6296928887356842470_VEBTi @ A2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) ) ) ).

% repli_emp
thf(fact_7854_tan__pi,axiom,
    ( ( tan_real @ pi )
    = zero_zero_real ) ).

% tan_pi
thf(fact_7855_tan__zero,axiom,
    ( ( tan_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% tan_zero
thf(fact_7856_tan__zero,axiom,
    ( ( tan_real @ zero_zero_real )
    = zero_zero_real ) ).

% tan_zero
thf(fact_7857_replicate__eq__replicate,axiom,
    ! [M: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M @ X )
        = ( replicate_VEBT_VEBT @ N @ Y ) )
      = ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_7858_tan__periodic__pi,axiom,
    ! [X: real] :
      ( ( tan_real @ ( plus_plus_real @ X @ pi ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_pi
thf(fact_7859_length__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_7860_length__replicate,axiom,
    ! [N: nat,X: real] :
      ( ( size_size_list_real @ ( replicate_real @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_7861_length__replicate,axiom,
    ! [N: nat,X: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_7862_length__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_7863_length__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( size_size_list_int @ ( replicate_int @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_7864_map__replicate,axiom,
    ! [F: nat > $o,N: nat,X: nat] :
      ( ( map_nat_o @ F @ ( replicate_nat @ N @ X ) )
      = ( replicate_o @ N @ ( F @ X ) ) ) ).

% map_replicate
thf(fact_7865_map__replicate,axiom,
    ! [F: nat > nat,N: nat,X: nat] :
      ( ( map_nat_nat @ F @ ( replicate_nat @ N @ X ) )
      = ( replicate_nat @ N @ ( F @ X ) ) ) ).

% map_replicate
thf(fact_7866_map__replicate,axiom,
    ! [F: vEBT_VEBT > real,N: nat,X: vEBT_VEBT] :
      ( ( map_VEBT_VEBT_real @ F @ ( replicate_VEBT_VEBT @ N @ X ) )
      = ( replicate_real @ N @ ( F @ X ) ) ) ).

% map_replicate
thf(fact_7867_map__replicate,axiom,
    ! [F: vEBT_VEBT > nat,N: nat,X: vEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat @ F @ ( replicate_VEBT_VEBT @ N @ X ) )
      = ( replicate_nat @ N @ ( F @ X ) ) ) ).

% map_replicate
thf(fact_7868_map__replicate,axiom,
    ! [F: vEBT_VEBT > vEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( map_VE8901447254227204932T_VEBT @ F @ ( replicate_VEBT_VEBT @ N @ X ) )
      = ( replicate_VEBT_VEBT @ N @ ( F @ X ) ) ) ).

% map_replicate
thf(fact_7869_Ball__set__replicate,axiom,
    ! [N: nat,A: real,P: real > $o] :
      ( ( ! [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ ( replicate_real @ N @ A ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_7870_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_7871_Ball__set__replicate,axiom,
    ! [N: nat,A: int,P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_7872_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
           => ( P @ X2 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_7873_Bex__set__replicate,axiom,
    ! [N: nat,A: real,P: real > $o] :
      ( ( ? [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ ( replicate_real @ N @ A ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_7874_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_7875_Bex__set__replicate,axiom,
    ! [N: nat,A: int,P: int > $o] :
      ( ( ? [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_7876_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
            & ( P @ X2 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_7877_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_7878_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_7879_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_7880_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_7881_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_7882_nth__replicate,axiom,
    ! [I: nat,N: nat,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_VEBT_VEBTi @ ( replicate_VEBT_VEBTi @ N @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_7883_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_7884_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_7885_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_7886_tan__npi,axiom,
    ! [N: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_7887_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_7888_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_7889_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_7890_set__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
        = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_replicate
thf(fact_7891_set__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( N != zero_zero_nat )
     => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
        = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% set_replicate
thf(fact_7892_set__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( N != zero_zero_nat )
     => ( ( set_int2 @ ( replicate_int @ N @ X ) )
        = ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% set_replicate
thf(fact_7893_set__replicate,axiom,
    ! [N: nat,X: real] :
      ( ( N != zero_zero_nat )
     => ( ( set_real2 @ ( replicate_real @ N @ X ) )
        = ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% set_replicate
thf(fact_7894_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_7895_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: $o > $o > heap_T8145700208782473153_VEBTi,Bnd: $o > $o > nat,F5: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A6: $o,B5: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A6 @ B5 ) )
         => ( time_T5737551269749752165_VEBTi @ ( F @ A6 @ B5 ) @ ( Bnd @ A6 @ B5 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
           => ( time_T5737551269749752165_VEBTi @ ( F5 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
       => ( time_T5737551269749752165_VEBTi @ ( vEBT_c6028912655521741485_VEBTi @ F5 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_7896_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: $o > $o > heap_Time_Heap_nat,Bnd: $o > $o > nat,F5: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_nat,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A6: $o,B5: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A6 @ B5 ) )
         => ( time_TBOUND_nat @ ( F @ A6 @ B5 ) @ ( Bnd @ A6 @ B5 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
           => ( time_TBOUND_nat @ ( F5 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
       => ( time_TBOUND_nat @ ( vEBT_c1335663792808957512ap_nat @ F5 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_7897_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: $o > $o > heap_T2636463487746394924on_nat,Bnd: $o > $o > nat,F5: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A6: $o,B5: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A6 @ B5 ) )
         => ( time_T8353473612707095248on_nat @ ( F @ A6 @ B5 ) @ ( Bnd @ A6 @ B5 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
           => ( time_T8353473612707095248on_nat @ ( F5 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
       => ( time_T8353473612707095248on_nat @ ( vEBT_c6250501799366334488on_nat @ F5 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_7898_Complex__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_0
thf(fact_7899_zero__complex_Ocode,axiom,
    ( zero_zero_complex
    = ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).

% zero_complex.code
thf(fact_7900_replicate__eqI,axiom,
    ! [Xs: list_complex,N: nat,X: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs )
        = N )
     => ( ! [Y3: complex] :
            ( ( member_complex @ Y3 @ ( set_complex2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_complex @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_7901_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = N )
     => ( ! [Y3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_VEBT_VEBT @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_7902_replicate__eqI,axiom,
    ! [Xs: list_real,N: nat,X: real] :
      ( ( ( size_size_list_real @ Xs )
        = N )
     => ( ! [Y3: real] :
            ( ( member_real @ Y3 @ ( set_real2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_real @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_7903_replicate__eqI,axiom,
    ! [Xs: list_o,N: nat,X: $o] :
      ( ( ( size_size_list_o @ Xs )
        = N )
     => ( ! [Y3: $o] :
            ( ( member_o @ Y3 @ ( set_o2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_o @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_7904_replicate__eqI,axiom,
    ! [Xs: list_nat,N: nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = N )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_7905_replicate__eqI,axiom,
    ! [Xs: list_int,N: nat,X: int] :
      ( ( ( size_size_list_int @ Xs )
        = N )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ ( set_int2 @ Xs ) )
           => ( Y3 = X ) )
       => ( Xs
          = ( replicate_int @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_7906_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( X3 = X ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_7907_replicate__length__same,axiom,
    ! [Xs: list_real,X: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ( X3 = X ) )
     => ( ( replicate_real @ ( size_size_list_real @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_7908_replicate__length__same,axiom,
    ! [Xs: list_o,X: $o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
         => ( X3 = X ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_7909_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( X3 = X ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_7910_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
         => ( X3 = X ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_7911_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_7912_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_7913_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ( complex2 @ A @ B )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( A
          = ( numeral_numeral_real @ W ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
thf(fact_7914_map__replicate__const,axiom,
    ! [K: real,Lst: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_real
        @ ^ [X2: vEBT_VEBT] : K
        @ Lst )
      = ( replicate_real @ ( size_s6755466524823107622T_VEBT @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_7915_map__replicate__const,axiom,
    ! [K: nat,Lst: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat
        @ ^ [X2: vEBT_VEBT] : K
        @ Lst )
      = ( replicate_nat @ ( size_s6755466524823107622T_VEBT @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_7916_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_real] :
      ( ( map_real_VEBT_VEBT
        @ ^ [X2: real] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_real @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_7917_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_o] :
      ( ( map_o_VEBT_VEBT
        @ ^ [X2: $o] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_o @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_7918_map__replicate__const,axiom,
    ! [K: $o,Lst: list_nat] :
      ( ( map_nat_o
        @ ^ [X2: nat] : K
        @ Lst )
      = ( replicate_o @ ( size_size_list_nat @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_7919_map__replicate__const,axiom,
    ! [K: nat,Lst: list_nat] :
      ( ( map_nat_nat
        @ ^ [X2: nat] : K
        @ Lst )
      = ( replicate_nat @ ( size_size_list_nat @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_7920_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_nat] :
      ( ( map_nat_VEBT_VEBT
        @ ^ [X2: nat] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_nat @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_7921_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_int] :
      ( ( map_int_VEBT_VEBT
        @ ^ [X2: int] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_int @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_7922_complex__add,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D2 ) )
      = ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D2 ) ) ) ).

% complex_add
thf(fact_7923_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_7924_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( A
          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
thf(fact_7925_complex__mult,axiom,
    ! [A: real,B: real,C: real,D2: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D2 ) )
      = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D2 ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D2 ) @ ( times_times_real @ B @ C ) ) ) ) ).

% complex_mult
thf(fact_7926_set__replicate__Suc,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N ) @ X ) )
      = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ).

% set_replicate_Suc
thf(fact_7927_set__replicate__Suc,axiom,
    ! [N: nat,X: nat] :
      ( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X ) )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% set_replicate_Suc
thf(fact_7928_set__replicate__Suc,axiom,
    ! [N: nat,X: int] :
      ( ( set_int2 @ ( replicate_int @ ( suc @ N ) @ X ) )
      = ( insert_int @ X @ bot_bot_set_int ) ) ).

% set_replicate_Suc
thf(fact_7929_set__replicate__Suc,axiom,
    ! [N: nat,X: real] :
      ( ( set_real2 @ ( replicate_real @ ( suc @ N ) @ X ) )
      = ( insert_real @ X @ bot_bot_set_real ) ) ).

% set_replicate_Suc
thf(fact_7930_tan__def,axiom,
    ( tan_complex
    = ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X2 ) @ ( cos_complex @ X2 ) ) ) ) ).

% tan_def
thf(fact_7931_tan__def,axiom,
    ( tan_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ X2 ) @ ( cos_real @ X2 ) ) ) ) ).

% tan_def
thf(fact_7932_set__replicate__conv__if,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
          = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_replicate_conv_if
thf(fact_7933_set__replicate__conv__if,axiom,
    ! [N: nat,X: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
          = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_7934_set__replicate__conv__if,axiom,
    ! [N: nat,X: int] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N @ X ) )
          = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N @ X ) )
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% set_replicate_conv_if
thf(fact_7935_set__replicate__conv__if,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N @ X ) )
          = bot_bot_set_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N @ X ) )
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% set_replicate_conv_if
thf(fact_7936_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_7937_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_7938_lemma__tan__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y @ ( tan_real @ X3 ) ) ) ) ).

% lemma_tan_total
thf(fact_7939_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X3 )
        = Y ) ) ).

% lemma_tan_total1
thf(fact_7940_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_7941_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_7942_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_7943_tan__total,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X3 )
        = Y )
      & ! [Y4: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
            & ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y4 )
              = Y ) )
         => ( Y4 = X3 ) ) ) ).

% tan_total
thf(fact_7944_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_7945_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_7946_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_7947_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_7948_tan__total__pos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X3 )
            = Y ) ) ) ).

% tan_total_pos
thf(fact_7949_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_7950_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_7951_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_7952_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_7953_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_7954_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_7955_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_7956_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_7957_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 ) )
     => ( ! [K2: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L4 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% and_int.pinduct
thf(fact_7958_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_7959_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_7960_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_7961_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_7962_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_7963_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_7964_tan__total__pi4,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ? [Z6: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z6 )
          & ( ord_less_real @ Z6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z6 )
            = X ) ) ) ).

% tan_total_pi4
thf(fact_7965_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.simps(3)
thf(fact_7966_tan__half,axiom,
    ( tan_complex
    = ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_complex ) ) ) ) ).

% tan_half
thf(fact_7967_tan__half,axiom,
    ( tan_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ) ).

% tan_half
thf(fact_7968_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I3: int,J: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J ) )
           => ( ( ( ord_less_eq_int @ I3 @ J )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J ) )
             => ( P @ I3 @ J ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_7969_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > assn,Y: real > assn] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X @ I2 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != one_one_assn ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( times_times_assn @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7970_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > assn,Y: vEBT_VEBT > assn] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
              & ( ( X @ I2 )
               != one_one_assn ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != one_one_assn ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( times_times_assn @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7971_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > assn,Y: nat > assn] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X @ I2 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != one_one_assn ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( times_times_assn @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7972_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > assn,Y: int > assn] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I2: int] :
              ( ( member_int @ I2 @ I5 )
              & ( ( X @ I2 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != one_one_assn ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( times_times_assn @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7973_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > assn,Y: complex > assn] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
              & ( ( X @ I2 )
               != one_one_assn ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != one_one_assn ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( times_times_assn @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7974_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( times_times_real @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7975_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > real,Y: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
              & ( ( X @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != one_one_real ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( times_times_real @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7976_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( times_times_real @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7977_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I2: int] :
              ( ( member_int @ I2 @ I5 )
              & ( ( X @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( times_times_real @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7978_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
              & ( ( X @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != one_one_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( times_times_real @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_7979_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
            & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_7980_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > complex,Y: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X @ I2 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( plus_plus_complex @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7981_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > complex,Y: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
              & ( ( X @ I2 )
               != zero_zero_complex ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != zero_zero_complex ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( plus_plus_complex @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7982_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X @ I2 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( plus_plus_complex @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7983_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I2: int] :
              ( ( member_int @ I2 @ I5 )
              & ( ( X @ I2 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( plus_plus_complex @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7984_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
              & ( ( X @ I2 )
               != zero_zero_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != zero_zero_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( plus_plus_complex @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7985_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X @ I2 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( plus_plus_real @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7986_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > real,Y: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
              & ( ( X @ I2 )
               != zero_zero_real ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != zero_zero_real ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
                & ( ( plus_plus_real @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7987_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X @ I2 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( plus_plus_real @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7988_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I2: int] :
              ( ( member_int @ I2 @ I5 )
              & ( ( X @ I2 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( plus_plus_real @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7989_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
              & ( ( X @ I2 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( Y @ I2 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( plus_plus_real @ ( X @ I2 ) @ ( Y @ I2 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_7990_sum__gp,axiom,
    ! [N: nat,M: nat,X: rat] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7991_sum__gp,axiom,
    ! [N: nat,M: nat,X: complex] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7992_sum__gp,axiom,
    ! [N: nat,M: nat,X: real] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7993_powr__0,axiom,
    ! [Z: real] :
      ( ( powr_real @ zero_zero_real @ Z )
      = zero_zero_real ) ).

% powr_0
thf(fact_7994_powr__eq__0__iff,axiom,
    ! [W: real,Z: real] :
      ( ( ( powr_real @ W @ Z )
        = zero_zero_real )
      = ( W = zero_zero_real ) ) ).

% powr_eq_0_iff
thf(fact_7995_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A2 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_7996_sum_Oneutral__const,axiom,
    ! [A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu3: complex] : zero_zero_complex
        @ A2 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_7997_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A2 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_7998_sum_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu3: int] : zero_zero_int
        @ A2 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_7999_of__nat__sum,axiom,
    ! [F: complex > nat,A2: set_complex] :
      ( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( semiri8010041392384452111omplex @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_8000_of__nat__sum,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X2: int] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_8001_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_8002_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri8010041392384452111omplex @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [X2: nat] : ( semiri8010041392384452111omplex @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_8003_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_8004_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_8005_of__int__sum,axiom,
    ! [F: complex > int,A2: set_complex] :
      ( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( ring_17405671764205052669omplex @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_8006_of__int__sum,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( ring_1_of_int_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_8007_of__int__sum,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_real @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups8778361861064173332t_real
        @ ^ [X2: int] : ( ring_1_of_int_real @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_8008_of__int__sum,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups3906332499630173760nt_rat
        @ ^ [X2: int] : ( ring_1_of_int_rat @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_8009_of__int__sum,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_17405671764205052669omplex @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups3049146728041665814omplex
        @ ^ [X2: int] : ( ring_17405671764205052669omplex @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_8010_of__int__sum,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X2: int] : ( ring_1_of_int_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_8011_sum_Oempty,axiom,
    ! [G: nat > complex] :
      ( ( groups2073611262835488442omplex @ G @ bot_bot_set_nat )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_8012_sum_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_8013_sum_Oempty,axiom,
    ! [G: nat > int] :
      ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
      = zero_zero_int ) ).

% sum.empty
thf(fact_8014_sum_Oempty,axiom,
    ! [G: nat > code_integer] :
      ( ( groups7501900531339628137nteger @ G @ bot_bot_set_nat )
      = zero_z3403309356797280102nteger ) ).

% sum.empty
thf(fact_8015_sum_Oempty,axiom,
    ! [G: int > complex] :
      ( ( groups3049146728041665814omplex @ G @ bot_bot_set_int )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_8016_sum_Oempty,axiom,
    ! [G: int > real] :
      ( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
      = zero_zero_real ) ).

% sum.empty
thf(fact_8017_sum_Oempty,axiom,
    ! [G: int > rat] :
      ( ( groups3906332499630173760nt_rat @ G @ bot_bot_set_int )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_8018_sum_Oempty,axiom,
    ! [G: int > nat] :
      ( ( groups4541462559716669496nt_nat @ G @ bot_bot_set_int )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_8019_sum_Oempty,axiom,
    ! [G: int > code_integer] :
      ( ( groups7873554091576472773nteger @ G @ bot_bot_set_int )
      = zero_z3403309356797280102nteger ) ).

% sum.empty
thf(fact_8020_sum_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_8021_sum__eq__0__iff,axiom,
    ! [F2: set_int,F: int > nat] :
      ( ( finite_finite_int @ F2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X2: int] :
              ( ( member_int @ X2 @ F2 )
             => ( ( F @ X2 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_8022_sum__eq__0__iff,axiom,
    ! [F2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ F2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X2: complex] :
              ( ( member_complex @ X2 @ F2 )
             => ( ( F @ X2 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_8023_sum__eq__0__iff,axiom,
    ! [F2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ F2 )
          = zero_zero_nat )
        = ( ! [X2: nat] :
              ( ( member_nat @ X2 @ F2 )
             => ( ( F @ X2 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_8024_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > complex] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups2073611262835488442omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_8025_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > complex] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_8026_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_8027_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_8028_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_8029_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_8030_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_8031_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_8032_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_8033_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > int] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups3539618377306564664at_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.infinite
thf(fact_8034_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_8035_powr__gt__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
      = ( X != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_8036_powr__nonneg__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_8037_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_8038_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_8039_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups1794756597179926696omplex
              @ ^ [K3: vEBT_VEBT] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups1794756597179926696omplex
              @ ^ [K3: vEBT_VEBT] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_8040_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_8041_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_8042_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_8043_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_8044_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_8045_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_8046_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_8047_sum_Odelta,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_8048_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_8049_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups1794756597179926696omplex
              @ ^ [K3: vEBT_VEBT] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups1794756597179926696omplex
              @ ^ [K3: vEBT_VEBT] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_8050_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_8051_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_8052_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_8053_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_8054_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_8055_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_8056_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_8057_sum_Odelta_H,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_8058_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_8059_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_8060_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_8061_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_8062_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_8063_sum_Oinsert,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ~ ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_8064_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_8065_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_8066_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_8067_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_8068_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_8069_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_8070_powr__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ one_one_real )
        = X ) ) ).

% powr_one
thf(fact_8071_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_8072_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] :
      ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_powr_numeral_real
thf(fact_8073_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( abs_abs_real @ ( F @ I2 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_8074_sum__abs__ge__zero,axiom,
    ! [F: int > int,A2: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I2: int] : ( abs_abs_int @ ( F @ I2 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_8075_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_8076_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_8077_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_8078_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_8079_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_8080_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = ( plus_p5714425477246183910nteger @ ( groups7501900531339628137nteger @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_8081_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_8082_sum_Oop__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_8083_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8084_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8085_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8086_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_p5714425477246183910nteger @ ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8087_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8088_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8089_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_8090_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I2: nat] : ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I2: nat] : ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_8091_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_8092_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_8093_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > complex,D2: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) ) @ ( D2 @ I2 ) )
            @ A2 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D2 @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) ) @ ( D2 @ I2 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_8094_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > rat,D2: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) ) @ ( D2 @ I2 ) )
            @ A2 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D2 @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) ) @ ( D2 @ I2 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_8095_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > real,D2: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) ) @ ( D2 @ I2 ) )
            @ A2 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D2 @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) ) @ ( D2 @ I2 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_8096_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_8097_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_8098_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( plus_plus_complex @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_8099_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X2: nat] : ( plus_plus_real @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_8100_sum_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X2: int] : ( plus_plus_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
        @ A2 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_8101_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A2 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_8102_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A2 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A2 )
          = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_8103_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8104_sum__nonneg,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8105_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8106_sum__nonneg,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8107_sum__nonneg,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8108_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > code_integer] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( groups7501900531339628137nteger @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8109_sum__nonneg,axiom,
    ! [A2: set_real,F: real > code_integer] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( groups7713935264441627589nteger @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8110_sum__nonneg,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > code_integer] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( groups5748017345553531991nteger @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8111_sum__nonneg,axiom,
    ! [A2: set_int,F: int > code_integer] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( groups7873554091576472773nteger @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8112_sum__nonneg,axiom,
    ! [A2: set_complex,F: complex > code_integer] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( groups6621422865394947399nteger @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_8113_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8114_sum__nonpos,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8115_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8116_sum__nonpos,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8117_sum__nonpos,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8118_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > code_integer] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ zero_z3403309356797280102nteger ) )
     => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ A2 ) @ zero_z3403309356797280102nteger ) ) ).

% sum_nonpos
thf(fact_8119_sum__nonpos,axiom,
    ! [A2: set_real,F: real > code_integer] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A2 )
         => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ zero_z3403309356797280102nteger ) )
     => ( ord_le3102999989581377725nteger @ ( groups7713935264441627589nteger @ F @ A2 ) @ zero_z3403309356797280102nteger ) ) ).

% sum_nonpos
thf(fact_8120_sum__nonpos,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > code_integer] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A2 )
         => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ zero_z3403309356797280102nteger ) )
     => ( ord_le3102999989581377725nteger @ ( groups5748017345553531991nteger @ F @ A2 ) @ zero_z3403309356797280102nteger ) ) ).

% sum_nonpos
thf(fact_8121_sum__nonpos,axiom,
    ! [A2: set_int,F: int > code_integer] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ zero_z3403309356797280102nteger ) )
     => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ A2 ) @ zero_z3403309356797280102nteger ) ) ).

% sum_nonpos
thf(fact_8122_sum__nonpos,axiom,
    ! [A2: set_complex,F: complex > code_integer] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ zero_z3403309356797280102nteger ) )
     => ( ord_le3102999989581377725nteger @ ( groups6621422865394947399nteger @ F @ A2 ) @ zero_z3403309356797280102nteger ) ) ).

% sum_nonpos
thf(fact_8123_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > complex,A2: set_nat] :
      ( ( ( groups2073611262835488442omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A6: nat] :
            ( ( member_nat @ A6 @ A2 )
           => ( ( G @ A6 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8124_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > complex,A2: set_real] :
      ( ( ( groups5754745047067104278omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A6: real] :
            ( ( member_real @ A6 @ A2 )
           => ( ( G @ A6 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8125_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > complex,A2: set_VEBT_VEBT] :
      ( ( ( groups1794756597179926696omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A6: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A6 @ A2 )
           => ( ( G @ A6 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8126_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > complex,A2: set_int] :
      ( ( ( groups3049146728041665814omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A6: int] :
            ( ( member_int @ A6 @ A2 )
           => ( ( G @ A6 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8127_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A2: set_real] :
      ( ( ( groups8097168146408367636l_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A6: real] :
            ( ( member_real @ A6 @ A2 )
           => ( ( G @ A6 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8128_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > real,A2: set_VEBT_VEBT] :
      ( ( ( groups2240296850493347238T_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A6: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A6 @ A2 )
           => ( ( G @ A6 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8129_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A2: set_int] :
      ( ( ( groups8778361861064173332t_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A6: int] :
            ( ( member_int @ A6 @ A2 )
           => ( ( G @ A6 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8130_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > real,A2: set_complex] :
      ( ( ( groups5808333547571424918x_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A6: complex] :
            ( ( member_complex @ A6 @ A2 )
           => ( ( G @ A6 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8131_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A2: set_nat] :
      ( ( ( groups2906978787729119204at_rat @ G @ A2 )
       != zero_zero_rat )
     => ~ ! [A6: nat] :
            ( ( member_nat @ A6 @ A2 )
           => ( ( G @ A6 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8132_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > rat,A2: set_real] :
      ( ( ( groups1300246762558778688al_rat @ G @ A2 )
       != zero_zero_rat )
     => ~ ! [A6: real] :
            ( ( member_real @ A6 @ A2 )
           => ( ( G @ A6 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8133_sum_Oneutral,axiom,
    ! [A2: set_nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( G @ X3 )
            = zero_zero_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.neutral
thf(fact_8134_sum_Oneutral,axiom,
    ! [A2: set_complex,G: complex > complex] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A2 )
         => ( ( G @ X3 )
            = zero_zero_complex ) )
     => ( ( groups7754918857620584856omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.neutral
thf(fact_8135_sum_Oneutral,axiom,
    ! [A2: set_nat,G: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
         => ( ( G @ X3 )
            = zero_zero_real ) )
     => ( ( groups6591440286371151544t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.neutral
thf(fact_8136_sum_Oneutral,axiom,
    ! [A2: set_int,G: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A2 )
         => ( ( G @ X3 )
            = zero_zero_int ) )
     => ( ( groups4538972089207619220nt_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.neutral
thf(fact_8137_sum__divide__distrib,axiom,
    ! [F: complex > complex,A2: set_complex,R2: complex] :
      ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
      = ( groups7754918857620584856omplex
        @ ^ [N4: complex] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ R2 )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_8138_sum__divide__distrib,axiom,
    ! [F: nat > real,A2: set_nat,R2: real] :
      ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
      = ( groups6591440286371151544t_real
        @ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ R2 )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_8139_sum__product,axiom,
    ! [F: nat > nat,A2: set_nat,G: nat > nat,B3: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( times_times_nat @ ( F @ I2 ) @ ( G @ J3 ) )
            @ B3 )
        @ A2 ) ) ).

% sum_product
thf(fact_8140_sum__product,axiom,
    ! [F: complex > complex,A2: set_complex,G: complex > complex,B3: set_complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B3 ) )
      = ( groups7754918857620584856omplex
        @ ^ [I2: complex] :
            ( groups7754918857620584856omplex
            @ ^ [J3: complex] : ( times_times_complex @ ( F @ I2 ) @ ( G @ J3 ) )
            @ B3 )
        @ A2 ) ) ).

% sum_product
thf(fact_8141_sum__product,axiom,
    ! [F: nat > real,A2: set_nat,G: nat > real,B3: set_nat] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B3 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] :
            ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( times_times_real @ ( F @ I2 ) @ ( G @ J3 ) )
            @ B3 )
        @ A2 ) ) ).

% sum_product
thf(fact_8142_sum__product,axiom,
    ! [F: int > int,A2: set_int,G: int > int,B3: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I2: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J3: int] : ( times_times_int @ ( F @ I2 ) @ ( G @ J3 ) )
            @ B3 )
        @ A2 ) ) ).

% sum_product
thf(fact_8143_sum__distrib__right,axiom,
    ! [F: nat > nat,A2: set_nat,R2: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R2 )
      = ( groups3542108847815614940at_nat
        @ ^ [N4: nat] : ( times_times_nat @ ( F @ N4 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_8144_sum__distrib__right,axiom,
    ! [F: complex > complex,A2: set_complex,R2: complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
      = ( groups7754918857620584856omplex
        @ ^ [N4: complex] : ( times_times_complex @ ( F @ N4 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_8145_sum__distrib__right,axiom,
    ! [F: nat > real,A2: set_nat,R2: real] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
      = ( groups6591440286371151544t_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_8146_sum__distrib__right,axiom,
    ! [F: int > int,A2: set_int,R2: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R2 )
      = ( groups4538972089207619220nt_int
        @ ^ [N4: int] : ( times_times_int @ ( F @ N4 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_8147_sum__distrib__left,axiom,
    ! [R2: nat,F: nat > nat,A2: set_nat] :
      ( ( times_times_nat @ R2 @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N4: nat] : ( times_times_nat @ R2 @ ( F @ N4 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_8148_sum__distrib__left,axiom,
    ! [R2: complex,F: complex > complex,A2: set_complex] :
      ( ( times_times_complex @ R2 @ ( groups7754918857620584856omplex @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [N4: complex] : ( times_times_complex @ R2 @ ( F @ N4 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_8149_sum__distrib__left,axiom,
    ! [R2: real,F: nat > real,A2: set_nat] :
      ( ( times_times_real @ R2 @ ( groups6591440286371151544t_real @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [N4: nat] : ( times_times_real @ R2 @ ( F @ N4 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_8150_sum__distrib__left,axiom,
    ! [R2: int,F: int > int,A2: set_int] :
      ( ( times_times_int @ R2 @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N4: int] : ( times_times_int @ R2 @ ( F @ N4 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_8151_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_8152_mod__sum__eq,axiom,
    ! [F: nat > nat,A: nat,A2: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups3542108847815614940at_nat
          @ ^ [I2: nat] : ( modulo_modulo_nat @ ( F @ I2 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_8153_mod__sum__eq,axiom,
    ! [F: int > int,A: int,A2: set_int] :
      ( ( modulo_modulo_int
        @ ( groups4538972089207619220nt_int
          @ ^ [I2: int] : ( modulo_modulo_int @ ( F @ I2 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_8154_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > complex,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups5754745047067104278omplex @ G
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups5754745047067104278omplex
          @ ^ [X2: real] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8155_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > complex,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups1794756597179926696omplex @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups1794756597179926696omplex
          @ ^ [X2: vEBT_VEBT] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8156_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > complex,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2073611262835488442omplex @ G
          @ ( collect_nat
            @ ^ [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups2073611262835488442omplex
          @ ^ [X2: nat] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8157_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > complex,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups3049146728041665814omplex
          @ ^ [X2: int] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8158_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X2: real] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8159_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > real,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups2240296850493347238T_real @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups2240296850493347238T_real
          @ ^ [X2: vEBT_VEBT] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8160_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( collect_int
            @ ^ [X2: int] :
                ( ( member_int @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X2: int] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8161_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( collect_complex
            @ ^ [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X2: complex] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8162_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > rat,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat @ G
          @ ( collect_real
            @ ^ [X2: real] :
                ( ( member_real @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups1300246762558778688al_rat
          @ ^ [X2: real] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8163_sum_Ointer__filter,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > rat,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups136491112297645522BT_rat @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X2 @ A2 )
                & ( P @ X2 ) ) ) )
        = ( groups136491112297645522BT_rat
          @ ^ [X2: vEBT_VEBT] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_8164_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8165_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8166_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8167_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: nat > rat,I: nat > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8168_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: int > rat,I: int > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8169_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > rat,I: complex > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8170_sum__le__included,axiom,
    ! [S: set_complex,T: set_nat,G: nat > rat,I: nat > complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8171_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: int > rat,I: int > complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8172_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > rat,I: complex > complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8173_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: nat > code_integer,I: nat > nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X3 )
                    & ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ S ) @ ( groups7501900531339628137nteger @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8174_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: real] :
                ( ( member_real @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8175_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8176_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8177_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8178_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8179_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > code_integer] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( ( groups7713935264441627589nteger @ F @ A2 )
            = zero_z3403309356797280102nteger )
          = ( ! [X2: real] :
                ( ( member_real @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8180_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > code_integer] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( ( groups5748017345553531991nteger @ F @ A2 )
            = zero_z3403309356797280102nteger )
          = ( ! [X2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8181_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( ( groups7501900531339628137nteger @ F @ A2 )
            = zero_z3403309356797280102nteger )
          = ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8182_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( ( groups7873554091576472773nteger @ F @ A2 )
            = zero_z3403309356797280102nteger )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8183_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( ( groups6621422865394947399nteger @ F @ A2 )
            = zero_z3403309356797280102nteger )
          = ( ! [X2: complex] :
                ( ( member_complex @ X2 @ A2 )
               => ( ( F @ X2 )
                  = zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8184_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8185_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8186_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8187_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > code_integer,G: nat > code_integer] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
              & ( ord_le6747313008572928689nteger @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( groups7501900531339628137nteger @ F @ A2 ) @ ( groups7501900531339628137nteger @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8188_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > code_integer,G: int > code_integer] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( ord_le6747313008572928689nteger @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( groups7873554091576472773nteger @ F @ A2 ) @ ( groups7873554091576472773nteger @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8189_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > code_integer,G: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_le3102999989581377725nteger @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( ord_le6747313008572928689nteger @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_le6747313008572928689nteger @ ( groups6621422865394947399nteger @ F @ A2 ) @ ( groups6621422865394947399nteger @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8190_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8191_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8192_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8193_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
              & ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8194_sum_Orelated,axiom,
    ! [R3: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R3 @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X15: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( ( R3 @ X15 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_complex @ X15 @ Y1 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups2073611262835488442omplex @ H2 @ S2 ) @ ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8195_sum_Orelated,axiom,
    ! [R3: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
      ( ( R3 @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X15: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( ( R3 @ X15 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_complex @ X15 @ Y1 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups3049146728041665814omplex @ H2 @ S2 ) @ ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8196_sum_Orelated,axiom,
    ! [R3: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
      ( ( R3 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R3 @ X15 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_real @ X15 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups8778361861064173332t_real @ H2 @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8197_sum_Orelated,axiom,
    ! [R3: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( R3 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R3 @ X15 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_real @ X15 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups5808333547571424918x_real @ H2 @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8198_sum_Orelated,axiom,
    ! [R3: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R3 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R3 @ X15 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_rat @ X15 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups2906978787729119204at_rat @ H2 @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8199_sum_Orelated,axiom,
    ! [R3: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
      ( ( R3 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R3 @ X15 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_rat @ X15 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups3906332499630173760nt_rat @ H2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8200_sum_Orelated,axiom,
    ! [R3: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R3 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R3 @ X15 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_rat @ X15 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups5058264527183730370ex_rat @ H2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8201_sum_Orelated,axiom,
    ! [R3: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
      ( ( R3 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( ( R3 @ X15 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_nat @ X15 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups4541462559716669496nt_nat @ H2 @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8202_sum_Orelated,axiom,
    ! [R3: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R3 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( ( R3 @ X15 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_nat @ X15 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups5693394587270226106ex_nat @ H2 @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8203_sum_Orelated,axiom,
    ! [R3: int > int > $o,S2: set_nat,H2: nat > int,G: nat > int] :
      ( ( R3 @ zero_zero_int @ zero_zero_int )
     => ( ! [X15: int,Y1: int,X23: int,Y23: int] :
            ( ( ( R3 @ X15 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_int @ X15 @ Y1 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ( R3 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R3 @ ( groups3539618377306564664at_int @ H2 @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_8204_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ ( groups2240296850493347238T_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8205_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8206_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8207_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8208_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ ( groups136491112297645522BT_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8209_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8210_sum__strict__mono,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8211_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8212_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A2 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8213_sum__strict__mono,axiom,
    ! [A2: set_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A2 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups771621172384141258BT_nat @ F @ A2 ) @ ( groups771621172384141258BT_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8214_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( groups8097168146408367636l_real @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8215_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( groups2240296850493347238T_real @ G @ A2 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8216_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
            = ( groups8778361861064173332t_real @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8217_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5808333547571424918x_real @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8218_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
            = ( groups1300246762558778688al_rat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8219_sum_Oinsert__if,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( groups136491112297645522BT_rat @ G @ A2 ) ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A2 )
         => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8220_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( groups2906978787729119204at_rat @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8221_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( groups3906332499630173760nt_rat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8222_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5058264527183730370ex_rat @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8223_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
            = ( groups1935376822645274424al_nat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8224_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T5: set_real,S2: set_real,I: real > real,J2: real > real,T6: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A6: real] :
              ( ( member_real @ A6 @ ( minus_minus_set_real @ S2 @ S4 ) )
             => ( ( I @ ( J2 @ A6 ) )
                = A6 ) )
         => ( ! [A6: real] :
                ( ( member_real @ A6 @ ( minus_minus_set_real @ S2 @ S4 ) )
               => ( member_real @ ( J2 @ A6 ) @ ( minus_minus_set_real @ T6 @ T5 ) ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ T6 @ T5 ) )
                 => ( ( J2 @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: real] :
                    ( ( member_real @ B5 @ ( minus_minus_set_real @ T6 @ T5 ) )
                   => ( member_real @ ( I @ B5 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
               => ( ! [A6: real] :
                      ( ( member_real @ A6 @ S4 )
                     => ( ( G @ A6 )
                        = zero_zero_complex ) )
                 => ( ! [B5: real] :
                        ( ( member_real @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A6: real] :
                          ( ( member_real @ A6 @ S2 )
                         => ( ( H2 @ ( J2 @ A6 ) )
                            = ( G @ A6 ) ) )
                     => ( ( groups5754745047067104278omplex @ G @ S2 )
                        = ( groups5754745047067104278omplex @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8225_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T5: set_VEBT_VEBT,S2: set_real,I: vEBT_VEBT > real,J2: real > vEBT_VEBT,T6: set_VEBT_VEBT,G: real > complex,H2: vEBT_VEBT > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite5795047828879050333T_VEBT @ T5 )
       => ( ! [A6: real] :
              ( ( member_real @ A6 @ ( minus_minus_set_real @ S2 @ S4 ) )
             => ( ( I @ ( J2 @ A6 ) )
                = A6 ) )
         => ( ! [A6: real] :
                ( ( member_real @ A6 @ ( minus_minus_set_real @ S2 @ S4 ) )
               => ( member_VEBT_VEBT @ ( J2 @ A6 ) @ ( minus_5127226145743854075T_VEBT @ T6 @ T5 ) ) )
           => ( ! [B5: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ T6 @ T5 ) )
                 => ( ( J2 @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ T6 @ T5 ) )
                   => ( member_real @ ( I @ B5 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
               => ( ! [A6: real] :
                      ( ( member_real @ A6 @ S4 )
                     => ( ( G @ A6 )
                        = zero_zero_complex ) )
                 => ( ! [B5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A6: real] :
                          ( ( member_real @ A6 @ S2 )
                         => ( ( H2 @ ( J2 @ A6 ) )
                            = ( G @ A6 ) ) )
                     => ( ( groups5754745047067104278omplex @ G @ S2 )
                        = ( groups1794756597179926696omplex @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8226_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T5: set_real,S2: set_VEBT_VEBT,I: real > vEBT_VEBT,J2: vEBT_VEBT > real,T6: set_real,G: vEBT_VEBT > complex,H2: real > complex] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A6: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A6 @ ( minus_5127226145743854075T_VEBT @ S2 @ S4 ) )
             => ( ( I @ ( J2 @ A6 ) )
                = A6 ) )
         => ( ! [A6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A6 @ ( minus_5127226145743854075T_VEBT @ S2 @ S4 ) )
               => ( member_real @ ( J2 @ A6 ) @ ( minus_minus_set_real @ T6 @ T5 ) ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ T6 @ T5 ) )
                 => ( ( J2 @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: real] :
                    ( ( member_real @ B5 @ ( minus_minus_set_real @ T6 @ T5 ) )
                   => ( member_VEBT_VEBT @ ( I @ B5 ) @ ( minus_5127226145743854075T_VEBT @ S2 @ S4 ) ) )
               => ( ! [A6: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A6 @ S4 )
                     => ( ( G @ A6 )
                        = zero_zero_complex ) )
                 => ( ! [B5: real] :
                        ( ( member_real @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A6: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A6 @ S2 )
                         => ( ( H2 @ ( J2 @ A6 ) )
                            = ( G @ A6 ) ) )
                     => ( ( groups1794756597179926696omplex @ G @ S2 )
                        = ( groups5754745047067104278omplex @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8227_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T5: set_VEBT_VEBT,S2: set_VEBT_VEBT,I: vEBT_VEBT > vEBT_VEBT,J2: vEBT_VEBT > vEBT_VEBT,T6: set_VEBT_VEBT,G: vEBT_VEBT > complex,H2: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( finite5795047828879050333T_VEBT @ T5 )
       => ( ! [A6: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A6 @ ( minus_5127226145743854075T_VEBT @ S2 @ S4 ) )
             => ( ( I @ ( J2 @ A6 ) )
                = A6 ) )
         => ( ! [A6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A6 @ ( minus_5127226145743854075T_VEBT @ S2 @ S4 ) )
               => ( member_VEBT_VEBT @ ( J2 @ A6 ) @ ( minus_5127226145743854075T_VEBT @ T6 @ T5 ) ) )
           => ( ! [B5: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ T6 @ T5 ) )
                 => ( ( J2 @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ T6 @ T5 ) )
                   => ( member_VEBT_VEBT @ ( I @ B5 ) @ ( minus_5127226145743854075T_VEBT @ S2 @ S4 ) ) )
               => ( ! [A6: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A6 @ S4 )
                     => ( ( G @ A6 )
                        = zero_zero_complex ) )
                 => ( ! [B5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A6: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A6 @ S2 )
                         => ( ( H2 @ ( J2 @ A6 ) )
                            = ( G @ A6 ) ) )
                     => ( ( groups1794756597179926696omplex @ G @ S2 )
                        = ( groups1794756597179926696omplex @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8228_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T5: set_int,S2: set_real,I: int > real,J2: real > int,T6: set_int,G: real > complex,H2: int > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A6: real] :
              ( ( member_real @ A6 @ ( minus_minus_set_real @ S2 @ S4 ) )
             => ( ( I @ ( J2 @ A6 ) )
                = A6 ) )
         => ( ! [A6: real] :
                ( ( member_real @ A6 @ ( minus_minus_set_real @ S2 @ S4 ) )
               => ( member_int @ ( J2 @ A6 ) @ ( minus_minus_set_int @ T6 @ T5 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T6 @ T5 ) )
                 => ( ( J2 @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T6 @ T5 ) )
                   => ( member_real @ ( I @ B5 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
               => ( ! [A6: real] :
                      ( ( member_real @ A6 @ S4 )
                     => ( ( G @ A6 )
                        = zero_zero_complex ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A6: real] :
                          ( ( member_real @ A6 @ S2 )
                         => ( ( H2 @ ( J2 @ A6 ) )
                            = ( G @ A6 ) ) )
                     => ( ( groups5754745047067104278omplex @ G @ S2 )
                        = ( groups3049146728041665814omplex @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8229_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_VEBT_VEBT,T5: set_int,S2: set_VEBT_VEBT,I: int > vEBT_VEBT,J2: vEBT_VEBT > int,T6: set_int,G: vEBT_VEBT > complex,H2: int > complex] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A6: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A6 @ ( minus_5127226145743854075T_VEBT @ S2 @ S4 ) )
             => ( ( I @ ( J2 @ A6 ) )
                = A6 ) )
         => ( ! [A6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A6 @ ( minus_5127226145743854075T_VEBT @ S2 @ S4 ) )
               => ( member_int @ ( J2 @ A6 ) @ ( minus_minus_set_int @ T6 @ T5 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T6 @ T5 ) )
                 => ( ( J2 @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T6 @ T5 ) )
                   => ( member_VEBT_VEBT @ ( I @ B5 ) @ ( minus_5127226145743854075T_VEBT @ S2 @ S4 ) ) )
               => ( ! [A6: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A6 @ S4 )
                     => ( ( G @ A6 )
                        = zero_zero_complex ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A6: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A6 @ S2 )
                         => ( ( H2 @ ( J2 @ A6 ) )
                            = ( G @ A6 ) ) )
                     => ( ( groups1794756597179926696omplex @ G @ S2 )
                        = ( groups3049146728041665814omplex @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8230_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T5: set_real,S2: set_int,I: real > int,J2: int > real,T6: set_real,G: int > complex,H2: real > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A6: int] :
              ( ( member_int @ A6 @ ( minus_minus_set_int @ S2 @ S4 ) )
             => ( ( I @ ( J2 @ A6 ) )
                = A6 ) )
         => ( ! [A6: int] :
                ( ( member_int @ A6 @ ( minus_minus_set_int @ S2 @ S4 ) )
               => ( member_real @ ( J2 @ A6 ) @ ( minus_minus_set_real @ T6 @ T5 ) ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ T6 @ T5 ) )
                 => ( ( J2 @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: real] :
                    ( ( member_real @ B5 @ ( minus_minus_set_real @ T6 @ T5 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
               => ( ! [A6: int] :
                      ( ( member_int @ A6 @ S4 )
                     => ( ( G @ A6 )
                        = zero_zero_complex ) )
                 => ( ! [B5: real] :
                        ( ( member_real @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A6: int] :
                          ( ( member_int @ A6 @ S2 )
                         => ( ( H2 @ ( J2 @ A6 ) )
                            = ( G @ A6 ) ) )
                     => ( ( groups3049146728041665814omplex @ G @ S2 )
                        = ( groups5754745047067104278omplex @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8231_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T5: set_VEBT_VEBT,S2: set_int,I: vEBT_VEBT > int,J2: int > vEBT_VEBT,T6: set_VEBT_VEBT,G: int > complex,H2: vEBT_VEBT > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite5795047828879050333T_VEBT @ T5 )
       => ( ! [A6: int] :
              ( ( member_int @ A6 @ ( minus_minus_set_int @ S2 @ S4 ) )
             => ( ( I @ ( J2 @ A6 ) )
                = A6 ) )
         => ( ! [A6: int] :
                ( ( member_int @ A6 @ ( minus_minus_set_int @ S2 @ S4 ) )
               => ( member_VEBT_VEBT @ ( J2 @ A6 ) @ ( minus_5127226145743854075T_VEBT @ T6 @ T5 ) ) )
           => ( ! [B5: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ T6 @ T5 ) )
                 => ( ( J2 @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ T6 @ T5 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
               => ( ! [A6: int] :
                      ( ( member_int @ A6 @ S4 )
                     => ( ( G @ A6 )
                        = zero_zero_complex ) )
                 => ( ! [B5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A6: int] :
                          ( ( member_int @ A6 @ S2 )
                         => ( ( H2 @ ( J2 @ A6 ) )
                            = ( G @ A6 ) ) )
                     => ( ( groups3049146728041665814omplex @ G @ S2 )
                        = ( groups1794756597179926696omplex @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8232_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T5: set_int,S2: set_int,I: int > int,J2: int > int,T6: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A6: int] :
              ( ( member_int @ A6 @ ( minus_minus_set_int @ S2 @ S4 ) )
             => ( ( I @ ( J2 @ A6 ) )
                = A6 ) )
         => ( ! [A6: int] :
                ( ( member_int @ A6 @ ( minus_minus_set_int @ S2 @ S4 ) )
               => ( member_int @ ( J2 @ A6 ) @ ( minus_minus_set_int @ T6 @ T5 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T6 @ T5 ) )
                 => ( ( J2 @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T6 @ T5 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
               => ( ! [A6: int] :
                      ( ( member_int @ A6 @ S4 )
                     => ( ( G @ A6 )
                        = zero_zero_complex ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A6: int] :
                          ( ( member_int @ A6 @ S2 )
                         => ( ( H2 @ ( J2 @ A6 ) )
                            = ( G @ A6 ) ) )
                     => ( ( groups3049146728041665814omplex @ G @ S2 )
                        = ( groups3049146728041665814omplex @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8233_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T5: set_real,S2: set_real,I: real > real,J2: real > real,T6: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A6: real] :
              ( ( member_real @ A6 @ ( minus_minus_set_real @ S2 @ S4 ) )
             => ( ( I @ ( J2 @ A6 ) )
                = A6 ) )
         => ( ! [A6: real] :
                ( ( member_real @ A6 @ ( minus_minus_set_real @ S2 @ S4 ) )
               => ( member_real @ ( J2 @ A6 ) @ ( minus_minus_set_real @ T6 @ T5 ) ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ T6 @ T5 ) )
                 => ( ( J2 @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: real] :
                    ( ( member_real @ B5 @ ( minus_minus_set_real @ T6 @ T5 ) )
                   => ( member_real @ ( I @ B5 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
               => ( ! [A6: real] :
                      ( ( member_real @ A6 @ S4 )
                     => ( ( G @ A6 )
                        = zero_zero_real ) )
                 => ( ! [B5: real] :
                        ( ( member_real @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A6: real] :
                          ( ( member_real @ A6 @ S2 )
                         => ( ( H2 @ ( J2 @ A6 ) )
                            = ( G @ A6 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S2 )
                        = ( groups8097168146408367636l_real @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8234_powr__non__neg,axiom,
    ! [A: real,X: real] :
      ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_8235_powr__less__mono2__neg,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_8236_powr__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_8237_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).

% powr_ge_pzero
thf(fact_8238_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_8239_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_8240_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8241_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > rat,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8242_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8243_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8244_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8245_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > code_integer,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups7713935264441627589nteger @ F @ S )
            = zero_z3403309356797280102nteger )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8246_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > code_integer,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I3 @ S )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups5748017345553531991nteger @ F @ S )
            = zero_z3403309356797280102nteger )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ( F @ I )
              = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8247_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > code_integer,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups7501900531339628137nteger @ F @ S )
            = zero_z3403309356797280102nteger )
         => ( ( member_nat @ I @ S )
           => ( ( F @ I )
              = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8248_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > code_integer,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups7873554091576472773nteger @ F @ S )
            = zero_z3403309356797280102nteger )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8249_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > code_integer,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups6621422865394947399nteger @ F @ S )
            = zero_z3403309356797280102nteger )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_z3403309356797280102nteger ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8250_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > rat,B3: rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = B3 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8251_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > rat,B3: rat,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ S )
            = B3 )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8252_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > rat,B3: rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = B3 )
         => ( ( member_nat @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8253_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > rat,B3: rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = B3 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8254_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > rat,B3: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = B3 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8255_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > code_integer,B3: code_integer,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups7713935264441627589nteger @ F @ S )
            = B3 )
         => ( ( member_real @ I @ S )
           => ( ord_le3102999989581377725nteger @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8256_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > code_integer,B3: code_integer,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I3 @ S )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups5748017345553531991nteger @ F @ S )
            = B3 )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ord_le3102999989581377725nteger @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8257_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > code_integer,B3: code_integer,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups7501900531339628137nteger @ F @ S )
            = B3 )
         => ( ( member_nat @ I @ S )
           => ( ord_le3102999989581377725nteger @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8258_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > code_integer,B3: code_integer,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups7873554091576472773nteger @ F @ S )
            = B3 )
         => ( ( member_int @ I @ S )
           => ( ord_le3102999989581377725nteger @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8259_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > code_integer,B3: code_integer,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
       => ( ( ( groups6621422865394947399nteger @ F @ S )
            = B3 )
         => ( ( member_complex @ I @ S )
           => ( ord_le3102999989581377725nteger @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8260_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > complex] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_complex ) ) ) )
        = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8261_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8262_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8263_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8264_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8265_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8266_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_nat ) ) ) )
        = ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8267_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_zero_int ) ) ) )
        = ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8268_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups7873554091576472773nteger @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X2: int] :
                  ( ( G @ X2 )
                  = zero_z3403309356797280102nteger ) ) ) )
        = ( groups7873554091576472773nteger @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8269_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups6621422865394947399nteger @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X2: complex] :
                  ( ( G @ X2 )
                  = zero_z3403309356797280102nteger ) ) ) )
        = ( groups6621422865394947399nteger @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8270_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).

% sum.shift_bounds_Suc_ivl
thf(fact_8271_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > real,M: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
        @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).

% sum.shift_bounds_Suc_ivl
thf(fact_8272_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_8273_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > real,M: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_8274_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
        @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).

% sum.shift_bounds_nat_ivl
thf(fact_8275_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > real,M: nat,K: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
        @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).

% sum.shift_bounds_nat_ivl
thf(fact_8276_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_8277_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > real,M: nat,K: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_8278_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8279_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( member_VEBT_VEBT @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8280_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8281_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8282_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8283_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > code_integer] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups7713935264441627589nteger @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8284_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: vEBT_VEBT > code_integer] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( member_VEBT_VEBT @ I @ I5 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I ) )
         => ( ! [I3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups5748017345553531991nteger @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8285_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups7501900531339628137nteger @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8286_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > code_integer] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups7873554091576472773nteger @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8287_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( groups6621422865394947399nteger @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8288_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8289_sum__pos,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8290_sum__pos,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8291_sum__pos,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8292_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8293_sum__pos,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8294_sum__pos,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8295_sum__pos,axiom,
    ! [I5: set_int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8296_sum__pos,axiom,
    ! [I5: set_real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8297_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups771621172384141258BT_nat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8298_sum_Omono__neutral__cong__right,axiom,
    ! [T6: set_real,S2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ T6 )
     => ( ( ord_less_eq_set_real @ S2 @ T6 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ T6 )
              = ( groups5754745047067104278omplex @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8299_sum_Omono__neutral__cong__right,axiom,
    ! [T6: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: vEBT_VEBT > complex,H2: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT @ T6 )
     => ( ( ord_le4337996190870823476T_VEBT @ S2 @ T6 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1794756597179926696omplex @ G @ T6 )
              = ( groups1794756597179926696omplex @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8300_sum_Omono__neutral__cong__right,axiom,
    ! [T6: set_int,S2: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups3049146728041665814omplex @ G @ T6 )
              = ( groups3049146728041665814omplex @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8301_sum_Omono__neutral__cong__right,axiom,
    ! [T6: set_real,S2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T6 )
     => ( ( ord_less_eq_set_real @ S2 @ T6 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T6 )
              = ( groups8097168146408367636l_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8302_sum_Omono__neutral__cong__right,axiom,
    ! [T6: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ T6 )
     => ( ( ord_le4337996190870823476T_VEBT @ S2 @ T6 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2240296850493347238T_real @ G @ T6 )
              = ( groups2240296850493347238T_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8303_sum_Omono__neutral__cong__right,axiom,
    ! [T6: set_int,S2: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ T6 )
              = ( groups8778361861064173332t_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8304_sum_Omono__neutral__cong__right,axiom,
    ! [T6: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T6 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T6 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T6 )
              = ( groups5808333547571424918x_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8305_sum_Omono__neutral__cong__right,axiom,
    ! [T6: set_real,S2: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T6 )
     => ( ( ord_less_eq_set_real @ S2 @ T6 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ T6 )
              = ( groups1300246762558778688al_rat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8306_sum_Omono__neutral__cong__right,axiom,
    ! [T6: set_VEBT_VEBT,S2: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ T6 )
     => ( ( ord_le4337996190870823476T_VEBT @ S2 @ T6 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups136491112297645522BT_rat @ G @ T6 )
              = ( groups136491112297645522BT_rat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8307_sum_Omono__neutral__cong__right,axiom,
    ! [T6: set_int,S2: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ T6 )
              = ( groups3906332499630173760nt_rat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8308_sum_Omono__neutral__cong__left,axiom,
    ! [T6: set_real,S2: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T6 )
     => ( ( ord_less_eq_set_real @ S2 @ T6 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T6 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ S2 )
              = ( groups5754745047067104278omplex @ H2 @ T6 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8309_sum_Omono__neutral__cong__left,axiom,
    ! [T6: set_VEBT_VEBT,S2: set_VEBT_VEBT,H2: vEBT_VEBT > complex,G: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT @ T6 )
     => ( ( ord_le4337996190870823476T_VEBT @ S2 @ T6 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T6 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1794756597179926696omplex @ G @ S2 )
              = ( groups1794756597179926696omplex @ H2 @ T6 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8310_sum_Omono__neutral__cong__left,axiom,
    ! [T6: set_int,S2: set_int,H2: int > complex,G: int > complex] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups3049146728041665814omplex @ G @ S2 )
              = ( groups3049146728041665814omplex @ H2 @ T6 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8311_sum_Omono__neutral__cong__left,axiom,
    ! [T6: set_real,S2: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T6 )
     => ( ( ord_less_eq_set_real @ S2 @ T6 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T6 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S2 )
              = ( groups8097168146408367636l_real @ H2 @ T6 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8312_sum_Omono__neutral__cong__left,axiom,
    ! [T6: set_VEBT_VEBT,S2: set_VEBT_VEBT,H2: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ T6 )
     => ( ( ord_le4337996190870823476T_VEBT @ S2 @ T6 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T6 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2240296850493347238T_real @ G @ S2 )
              = ( groups2240296850493347238T_real @ H2 @ T6 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8313_sum_Omono__neutral__cong__left,axiom,
    ! [T6: set_int,S2: set_int,H2: int > real,G: int > real] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ S2 )
              = ( groups8778361861064173332t_real @ H2 @ T6 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8314_sum_Omono__neutral__cong__left,axiom,
    ! [T6: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T6 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T6 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T6 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S2 )
              = ( groups5808333547571424918x_real @ H2 @ T6 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8315_sum_Omono__neutral__cong__left,axiom,
    ! [T6: set_real,S2: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T6 )
     => ( ( ord_less_eq_set_real @ S2 @ T6 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T6 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ S2 )
              = ( groups1300246762558778688al_rat @ H2 @ T6 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8316_sum_Omono__neutral__cong__left,axiom,
    ! [T6: set_VEBT_VEBT,S2: set_VEBT_VEBT,H2: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ T6 )
     => ( ( ord_le4337996190870823476T_VEBT @ S2 @ T6 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T6 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups136491112297645522BT_rat @ G @ S2 )
              = ( groups136491112297645522BT_rat @ H2 @ T6 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8317_sum_Omono__neutral__cong__left,axiom,
    ! [T6: set_int,S2: set_int,H2: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ S2 )
              = ( groups3906332499630173760nt_rat @ H2 @ T6 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8318_sum_Omono__neutral__right,axiom,
    ! [T6: set_int,S2: set_int,G: int > complex] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G @ T6 )
            = ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8319_sum_Omono__neutral__right,axiom,
    ! [T6: set_int,S2: set_int,G: int > real] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ T6 )
            = ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8320_sum_Omono__neutral__right,axiom,
    ! [T6: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T6 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T6 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T6 )
            = ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8321_sum_Omono__neutral__right,axiom,
    ! [T6: set_int,S2: set_int,G: int > rat] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ T6 )
            = ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8322_sum_Omono__neutral__right,axiom,
    ! [T6: set_complex,S2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T6 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T6 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ T6 )
            = ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8323_sum_Omono__neutral__right,axiom,
    ! [T6: set_int,S2: set_int,G: int > nat] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ T6 )
            = ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8324_sum_Omono__neutral__right,axiom,
    ! [T6: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T6 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T6 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T6 )
            = ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8325_sum_Omono__neutral__right,axiom,
    ! [T6: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T6 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T6 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T6 )
            = ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8326_sum_Omono__neutral__right,axiom,
    ! [T6: set_int,S2: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7873554091576472773nteger @ G @ T6 )
            = ( groups7873554091576472773nteger @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8327_sum_Omono__neutral__right,axiom,
    ! [T6: set_complex,S2: set_complex,G: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ T6 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T6 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups6621422865394947399nteger @ G @ T6 )
            = ( groups6621422865394947399nteger @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8328_sum_Omono__neutral__left,axiom,
    ! [T6: set_int,S2: set_int,G: int > complex] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G @ S2 )
            = ( groups3049146728041665814omplex @ G @ T6 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8329_sum_Omono__neutral__left,axiom,
    ! [T6: set_int,S2: set_int,G: int > real] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ S2 )
            = ( groups8778361861064173332t_real @ G @ T6 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8330_sum_Omono__neutral__left,axiom,
    ! [T6: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T6 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T6 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S2 )
            = ( groups5808333547571424918x_real @ G @ T6 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8331_sum_Omono__neutral__left,axiom,
    ! [T6: set_int,S2: set_int,G: int > rat] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ S2 )
            = ( groups3906332499630173760nt_rat @ G @ T6 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8332_sum_Omono__neutral__left,axiom,
    ! [T6: set_complex,S2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T6 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T6 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ S2 )
            = ( groups5058264527183730370ex_rat @ G @ T6 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8333_sum_Omono__neutral__left,axiom,
    ! [T6: set_int,S2: set_int,G: int > nat] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ S2 )
            = ( groups4541462559716669496nt_nat @ G @ T6 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8334_sum_Omono__neutral__left,axiom,
    ! [T6: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T6 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T6 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S2 )
            = ( groups5693394587270226106ex_nat @ G @ T6 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8335_sum_Omono__neutral__left,axiom,
    ! [T6: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T6 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T6 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S2 )
            = ( groups5690904116761175830ex_int @ G @ T6 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8336_sum_Omono__neutral__left,axiom,
    ! [T6: set_int,S2: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ T6 )
     => ( ( ord_less_eq_set_int @ S2 @ T6 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups7873554091576472773nteger @ G @ S2 )
            = ( groups7873554091576472773nteger @ G @ T6 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8337_sum_Omono__neutral__left,axiom,
    ! [T6: set_complex,S2: set_complex,G: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ T6 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T6 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T6 @ S2 ) )
             => ( ( G @ X3 )
                = zero_z3403309356797280102nteger ) )
         => ( ( groups6621422865394947399nteger @ G @ S2 )
            = ( groups6621422865394947399nteger @ G @ T6 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8338_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B3 @ C5 )
         => ( ! [A6: real] :
                ( ( member_real @ A6 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_complex ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ C5 )
                  = ( groups5754745047067104278omplex @ H2 @ C5 ) )
               => ( ( groups5754745047067104278omplex @ G @ A2 )
                  = ( groups5754745047067104278omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8339_sum_Osame__carrierI,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,G: vEBT_VEBT > complex,H2: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT @ C5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C5 )
       => ( ( ord_le4337996190870823476T_VEBT @ B3 @ C5 )
         => ( ! [A6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A6 @ ( minus_5127226145743854075T_VEBT @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_complex ) )
           => ( ! [B5: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups1794756597179926696omplex @ G @ C5 )
                  = ( groups1794756597179926696omplex @ H2 @ C5 ) )
               => ( ( groups1794756597179926696omplex @ G @ A2 )
                  = ( groups1794756597179926696omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8340_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B3 @ C5 )
         => ( ! [A6: int] :
                ( ( member_int @ A6 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_complex ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups3049146728041665814omplex @ G @ C5 )
                  = ( groups3049146728041665814omplex @ H2 @ C5 ) )
               => ( ( groups3049146728041665814omplex @ G @ A2 )
                  = ( groups3049146728041665814omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8341_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B3 @ C5 )
         => ( ! [A6: real] :
                ( ( member_real @ A6 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_real ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C5 )
                  = ( groups8097168146408367636l_real @ H2 @ C5 ) )
               => ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8342_sum_Osame__carrierI,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ C5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C5 )
       => ( ( ord_le4337996190870823476T_VEBT @ B3 @ C5 )
         => ( ! [A6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A6 @ ( minus_5127226145743854075T_VEBT @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_real ) )
           => ( ! [B5: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups2240296850493347238T_real @ G @ C5 )
                  = ( groups2240296850493347238T_real @ H2 @ C5 ) )
               => ( ( groups2240296850493347238T_real @ G @ A2 )
                  = ( groups2240296850493347238T_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8343_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B3 @ C5 )
         => ( ! [A6: int] :
                ( ( member_int @ A6 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_real ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ C5 )
                  = ( groups8778361861064173332t_real @ H2 @ C5 ) )
               => ( ( groups8778361861064173332t_real @ G @ A2 )
                  = ( groups8778361861064173332t_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8344_sum_Osame__carrierI,axiom,
    ! [C5: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C5 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C5 )
         => ( ! [A6: complex] :
                ( ( member_complex @ A6 @ ( minus_811609699411566653omplex @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_real ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C5 )
                  = ( groups5808333547571424918x_real @ H2 @ C5 ) )
               => ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8345_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B3 @ C5 )
         => ( ! [A6: real] :
                ( ( member_real @ A6 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_rat ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ C5 )
                  = ( groups1300246762558778688al_rat @ H2 @ C5 ) )
               => ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8346_sum_Osame__carrierI,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ C5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C5 )
       => ( ( ord_le4337996190870823476T_VEBT @ B3 @ C5 )
         => ( ! [A6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A6 @ ( minus_5127226145743854075T_VEBT @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_rat ) )
           => ( ! [B5: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups136491112297645522BT_rat @ G @ C5 )
                  = ( groups136491112297645522BT_rat @ H2 @ C5 ) )
               => ( ( groups136491112297645522BT_rat @ G @ A2 )
                  = ( groups136491112297645522BT_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8347_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B3: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B3 @ C5 )
         => ( ! [A6: int] :
                ( ( member_int @ A6 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_rat ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ C5 )
                  = ( groups3906332499630173760nt_rat @ H2 @ C5 ) )
               => ( ( groups3906332499630173760nt_rat @ G @ A2 )
                  = ( groups3906332499630173760nt_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8348_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B3 @ C5 )
         => ( ! [A6: real] :
                ( ( member_real @ A6 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_complex ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ A2 )
                  = ( groups5754745047067104278omplex @ H2 @ B3 ) )
                = ( ( groups5754745047067104278omplex @ G @ C5 )
                  = ( groups5754745047067104278omplex @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8349_sum_Osame__carrier,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,G: vEBT_VEBT > complex,H2: vEBT_VEBT > complex] :
      ( ( finite5795047828879050333T_VEBT @ C5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C5 )
       => ( ( ord_le4337996190870823476T_VEBT @ B3 @ C5 )
         => ( ! [A6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A6 @ ( minus_5127226145743854075T_VEBT @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_complex ) )
           => ( ! [B5: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups1794756597179926696omplex @ G @ A2 )
                  = ( groups1794756597179926696omplex @ H2 @ B3 ) )
                = ( ( groups1794756597179926696omplex @ G @ C5 )
                  = ( groups1794756597179926696omplex @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8350_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B3 @ C5 )
         => ( ! [A6: int] :
                ( ( member_int @ A6 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_complex ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups3049146728041665814omplex @ G @ A2 )
                  = ( groups3049146728041665814omplex @ H2 @ B3 ) )
                = ( ( groups3049146728041665814omplex @ G @ C5 )
                  = ( groups3049146728041665814omplex @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8351_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B3 @ C5 )
         => ( ! [A6: real] :
                ( ( member_real @ A6 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_real ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B3 ) )
                = ( ( groups8097168146408367636l_real @ G @ C5 )
                  = ( groups8097168146408367636l_real @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8352_sum_Osame__carrier,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ C5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C5 )
       => ( ( ord_le4337996190870823476T_VEBT @ B3 @ C5 )
         => ( ! [A6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A6 @ ( minus_5127226145743854075T_VEBT @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_real ) )
           => ( ! [B5: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups2240296850493347238T_real @ G @ A2 )
                  = ( groups2240296850493347238T_real @ H2 @ B3 ) )
                = ( ( groups2240296850493347238T_real @ G @ C5 )
                  = ( groups2240296850493347238T_real @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8353_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B3 @ C5 )
         => ( ! [A6: int] :
                ( ( member_int @ A6 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_real ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ A2 )
                  = ( groups8778361861064173332t_real @ H2 @ B3 ) )
                = ( ( groups8778361861064173332t_real @ G @ C5 )
                  = ( groups8778361861064173332t_real @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8354_sum_Osame__carrier,axiom,
    ! [C5: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C5 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C5 )
         => ( ! [A6: complex] :
                ( ( member_complex @ A6 @ ( minus_811609699411566653omplex @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_real ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B3 ) )
                = ( ( groups5808333547571424918x_real @ G @ C5 )
                  = ( groups5808333547571424918x_real @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8355_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B3 @ C5 )
         => ( ! [A6: real] :
                ( ( member_real @ A6 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_rat ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B3 ) )
                = ( ( groups1300246762558778688al_rat @ G @ C5 )
                  = ( groups1300246762558778688al_rat @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8356_sum_Osame__carrier,axiom,
    ! [C5: set_VEBT_VEBT,A2: set_VEBT_VEBT,B3: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ C5 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ C5 )
       => ( ( ord_le4337996190870823476T_VEBT @ B3 @ C5 )
         => ( ! [A6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A6 @ ( minus_5127226145743854075T_VEBT @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_rat ) )
           => ( ! [B5: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups136491112297645522BT_rat @ G @ A2 )
                  = ( groups136491112297645522BT_rat @ H2 @ B3 ) )
                = ( ( groups136491112297645522BT_rat @ G @ C5 )
                  = ( groups136491112297645522BT_rat @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8357_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B3: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B3 @ C5 )
         => ( ! [A6: int] :
                ( ( member_int @ A6 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A6 )
                  = zero_zero_rat ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ A2 )
                  = ( groups3906332499630173760nt_rat @ H2 @ B3 ) )
                = ( ( groups3906332499630173760nt_rat @ G @ C5 )
                  = ( groups3906332499630173760nt_rat @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8358_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ A2 )
          = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups8778361861064173332t_real @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8359_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B3 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5808333547571424918x_real @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8360_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups3906332499630173760nt_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8361_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B3 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5058264527183730370ex_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8362_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8363_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B3 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5693394587270226106ex_nat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8364_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B3 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5690904116761175830ex_int @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8365_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A2: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B3 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups2906978787729119204at_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8366_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A2: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B3 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ A2 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups3539618377306564664at_int @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8367_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A2: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups3542108847815614940at_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups3542108847815614940at_nat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8368_sum_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat,G: nat > nat,H2: nat > nat] :
      ( ( A = C )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( ( ord_less_eq_nat @ C @ X3 )
             => ( ( ord_less_nat @ X3 @ D2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) ) )
         => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups3542108847815614940at_nat @ H2 @ ( set_or4665077453230672383an_nat @ C @ D2 ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_8369_sum_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat,G: nat > real,H2: nat > real] :
      ( ( A = C )
     => ( ( B = D2 )
       => ( ! [X3: nat] :
              ( ( ord_less_eq_nat @ C @ X3 )
             => ( ( ord_less_nat @ X3 @ D2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) ) )
         => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups6591440286371151544t_real @ H2 @ ( set_or4665077453230672383an_nat @ C @ D2 ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_8370_sum_Oivl__cong,axiom,
    ! [A: int,C: int,B: int,D2: int,G: int > int,H2: int > int] :
      ( ( A = C )
     => ( ( B = D2 )
       => ( ! [X3: int] :
              ( ( ord_less_eq_int @ C @ X3 )
             => ( ( ord_less_int @ X3 @ D2 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) ) )
         => ( ( groups4538972089207619220nt_int @ G @ ( set_or4662586982721622107an_int @ A @ B ) )
            = ( groups4538972089207619220nt_int @ H2 @ ( set_or4662586982721622107an_int @ C @ D2 ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_8371_powr__mono2_H,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_8372_powr__less__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_8373_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_8374_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_8375_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_8376_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_8377_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_8378_powr__divide,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 @ ( divide_divide_real @ X @ Y ) @ A )
          = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_divide
thf(fact_8379_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_8380_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P4: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P4 )
       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
          = ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_8381_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P4: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P4 )
       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
          = ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_8382_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P4: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P4 )
       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
          = ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_8383_sum_OatLeastLessThan__concat,axiom,
    ! [M: nat,N: nat,P4: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P4 )
       => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) )
          = ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_8384_sum__diff__nat__ivl,axiom,
    ! [M: nat,N: nat,P4: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P4 )
       => ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ M @ N ) ) )
          = ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_8385_sum__diff__nat__ivl,axiom,
    ! [M: nat,N: nat,P4: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P4 )
       => ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M @ N ) ) )
          = ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_8386_sum__diff__nat__ivl,axiom,
    ! [M: nat,N: nat,P4: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ P4 )
       => ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ M @ P4 ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ M @ N ) ) )
          = ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ N @ P4 ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_8387_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_8388_log__base__powr,axiom,
    ! [A: real,B: real,X: real] :
      ( ( A != zero_zero_real )
     => ( ( log @ ( powr_real @ A @ B ) @ X )
        = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).

% log_base_powr
thf(fact_8389_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_8390_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_8391_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_8392_powr__diff,axiom,
    ! [W: real,Z1: real,Z2: real] :
      ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z2 ) )
      = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z2 ) ) ) ).

% powr_diff
thf(fact_8393_sum__power__add,axiom,
    ! [X: complex,M: nat,I5: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I2: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8394_sum__power__add,axiom,
    ! [X: code_integer,M: nat,I5: set_nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [I2: nat] : ( power_8256067586552552935nteger @ X @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ M ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8395_sum__power__add,axiom,
    ! [X: rat,M: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I2: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8396_sum__power__add,axiom,
    ! [X: int,M: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I2: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8397_sum__power__add,axiom,
    ! [X: real,M: nat,I5: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8398_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ! [B5: real] :
              ( ( member_real @ B5 @ ( minus_minus_set_real @ B3 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B5 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_8399_sum__mono2,axiom,
    ! [B3: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ B3 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B3 )
       => ( ! [B5: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ B3 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B5 ) ) )
         => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ ( groups136491112297645522BT_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_8400_sum__mono2,axiom,
    ! [B3: set_int,A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A2 @ B3 )
       => ( ! [B5: int] :
              ( ( member_int @ B5 @ ( minus_minus_set_int @ B3 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B5 ) ) )
         => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_8401_sum__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ! [B5: complex] :
              ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B5 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_8402_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: real > code_integer] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ! [B5: real] :
              ( ( member_real @ B5 @ ( minus_minus_set_real @ B3 @ A2 ) )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ B5 ) ) )
         => ( ord_le3102999989581377725nteger @ ( groups7713935264441627589nteger @ F @ A2 ) @ ( groups7713935264441627589nteger @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_8403_sum__mono2,axiom,
    ! [B3: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > code_integer] :
      ( ( finite5795047828879050333T_VEBT @ B3 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B3 )
       => ( ! [B5: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ B3 @ A2 ) )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ B5 ) ) )
         => ( ord_le3102999989581377725nteger @ ( groups5748017345553531991nteger @ F @ A2 ) @ ( groups5748017345553531991nteger @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_8404_sum__mono2,axiom,
    ! [B3: set_int,A2: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A2 @ B3 )
       => ( ! [B5: int] :
              ( ( member_int @ B5 @ ( minus_minus_set_int @ B3 @ A2 ) )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ B5 ) ) )
         => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ F @ A2 ) @ ( groups7873554091576472773nteger @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_8405_sum__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ! [B5: complex] :
              ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
             => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ B5 ) ) )
         => ( ord_le3102999989581377725nteger @ ( groups6621422865394947399nteger @ F @ A2 ) @ ( groups6621422865394947399nteger @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_8406_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ! [B5: real] :
              ( ( member_real @ B5 @ ( minus_minus_set_real @ B3 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B5 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_8407_sum__mono2,axiom,
    ! [B3: set_VEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ B3 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B3 )
       => ( ! [B5: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B5 @ ( minus_5127226145743854075T_VEBT @ B3 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B5 ) ) )
         => ( ord_less_eq_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ ( groups2240296850493347238T_real @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_8408_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_8409_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_8410_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups2240296850493347238T_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8411_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8412_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups136491112297645522BT_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8413_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8414_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups771621172384141258BT_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups771621172384141258BT_nat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8415_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8416_sum_Oremove,axiom,
    ! [A2: set_VEBT_VEBT,X: vEBT_VEBT,G: vEBT_VEBT > int] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ( groups769130701875090982BT_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups769130701875090982BT_int @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8417_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8418_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8419_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8420_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > real,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups2240296850493347238T_real @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups2240296850493347238T_real @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8421_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > real,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8422_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > rat,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups136491112297645522BT_rat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups136491112297645522BT_rat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8423_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > rat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8424_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > nat,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups771621172384141258BT_nat @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups771621172384141258BT_nat @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8425_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > nat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8426_sum_Oinsert__remove,axiom,
    ! [A2: set_VEBT_VEBT,G: vEBT_VEBT > int,X: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( groups769130701875090982BT_int @ G @ ( insert_VEBT_VEBT @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups769130701875090982BT_int @ G @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8427_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > int,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8428_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > real,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8429_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > rat,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8430_suminf__finite,axiom,
    ! [N5: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_complex ) )
       => ( ( suminf_complex @ F )
          = ( groups2073611262835488442omplex @ F @ N5 ) ) ) ) ).

% suminf_finite
thf(fact_8431_suminf__finite,axiom,
    ! [N5: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_int ) )
       => ( ( suminf_int @ F )
          = ( groups3539618377306564664at_int @ F @ N5 ) ) ) ) ).

% suminf_finite
thf(fact_8432_suminf__finite,axiom,
    ! [N5: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_nat ) )
       => ( ( suminf_nat @ F )
          = ( groups3542108847815614940at_nat @ F @ N5 ) ) ) ) ).

% suminf_finite
thf(fact_8433_suminf__finite,axiom,
    ! [N5: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N5 )
     => ( ! [N2: nat] :
            ( ~ ( member_nat @ N2 @ N5 )
           => ( ( F @ N2 )
              = zero_zero_real ) )
       => ( ( suminf_real @ F )
          = ( groups6591440286371151544t_real @ F @ N5 ) ) ) ) ).

% suminf_finite
thf(fact_8434_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > real,C: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups2240296850493347238T_real @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups2240296850493347238T_real @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8435_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8436_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > rat,C: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups136491112297645522BT_rat @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups136491112297645522BT_rat @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8437_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: complex > rat,C: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8438_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > nat,C: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups771621172384141258BT_nat
              @ ^ [K3: vEBT_VEBT] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups771621172384141258BT_nat @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups771621172384141258BT_nat
              @ ^ [K3: vEBT_VEBT] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups771621172384141258BT_nat @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8439_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8440_sum_Odelta__remove,axiom,
    ! [S2: set_VEBT_VEBT,A: vEBT_VEBT,B: vEBT_VEBT > int,C: vEBT_VEBT > int] :
      ( ( finite5795047828879050333T_VEBT @ S2 )
     => ( ( ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups769130701875090982BT_int
              @ ^ [K3: vEBT_VEBT] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups769130701875090982BT_int @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A @ S2 )
         => ( ( groups769130701875090982BT_int
              @ ^ [K3: vEBT_VEBT] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups769130701875090982BT_int @ C @ ( minus_5127226145743854075T_VEBT @ S2 @ ( insert_VEBT_VEBT @ A @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8441_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: complex > int,C: complex > int] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8442_sum_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: int > real,C: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8443_sum_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: int > rat,C: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S2 )
            = ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8444_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( groups2073611262835488442omplex @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2073611262835488442omplex @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_8445_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > rat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_rat )
     => ( ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_8446_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > int,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_8447_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > code_integer,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_z3403309356797280102nteger )
     => ( ( groups7501900531339628137nteger @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups7501900531339628137nteger @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_8448_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > nat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_8449_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > real,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups6591440286371151544t_real @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_8450_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8451_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > rat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_rat )
     => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8452_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > int,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8453_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > code_integer,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_z3403309356797280102nteger )
     => ( ( groups7501900531339628137nteger @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups7501900531339628137nteger @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8454_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > nat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8455_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > real,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8456_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_8457_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_8458_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_8459_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_8460_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_8461_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_nat @ M @ N )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_8462_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_8463_sum_OatLeast__Suc__lessThan,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_nat @ M @ N )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_8464_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_8465_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_8466_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_8467_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_8468_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_8469_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_8470_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_8471_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_8472_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
        = ( power_power_real @ X @ N ) ) ) ).

% powr_realpow
thf(fact_8473_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8474_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8475_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8476_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8477_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8478_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8479_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8480_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8481_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_8482_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_8483_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_8484_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_8485_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_rat @ ( G @ N ) @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).

% sum.last_plus
thf(fact_8486_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_int @ ( G @ N ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).

% sum.last_plus
thf(fact_8487_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_nat @ ( G @ N ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).

% sum.last_plus
thf(fact_8488_sum_Olast__plus,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_real @ ( G @ N ) @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ) ).

% sum.last_plus
thf(fact_8489_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: real > rat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8490_sum__strict__mono2,axiom,
    ! [B3: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ B3 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B3 )
       => ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B3 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups136491112297645522BT_rat @ F @ A2 ) @ ( groups136491112297645522BT_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8491_sum__strict__mono2,axiom,
    ! [B3: set_int,A2: set_int,B: int,F: int > rat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A2 @ B3 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8492_sum__strict__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,B: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8493_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: real > code_integer] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B3 )
                 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
             => ( ord_le6747313008572928689nteger @ ( groups7713935264441627589nteger @ F @ A2 ) @ ( groups7713935264441627589nteger @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8494_sum__strict__mono2,axiom,
    ! [B3: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: vEBT_VEBT > code_integer] :
      ( ( finite5795047828879050333T_VEBT @ B3 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B3 )
       => ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B3 @ A2 ) )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ B ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B3 )
                 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
             => ( ord_le6747313008572928689nteger @ ( groups5748017345553531991nteger @ F @ A2 ) @ ( groups5748017345553531991nteger @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8495_sum__strict__mono2,axiom,
    ! [B3: set_int,A2: set_int,B: int,F: int > code_integer] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A2 @ B3 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A2 ) )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ B ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ B3 )
                 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
             => ( ord_le6747313008572928689nteger @ ( groups7873554091576472773nteger @ F @ A2 ) @ ( groups7873554091576472773nteger @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8496_sum__strict__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,B: complex,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( F @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B3 )
                 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
             => ( ord_le6747313008572928689nteger @ ( groups6621422865394947399nteger @ F @ A2 ) @ ( groups6621422865394947399nteger @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8497_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: real > real] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B3 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8498_sum__strict__mono2,axiom,
    ! [B3: set_VEBT_VEBT,A2: set_VEBT_VEBT,B: vEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ B3 )
     => ( ( ord_le4337996190870823476T_VEBT @ A2 @ B3 )
       => ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B3 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B3 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups2240296850493347238T_real @ F @ A2 ) @ ( groups2240296850493347238T_real @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8499_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( member_VEBT_VEBT @ I @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite5795047828879050333T_VEBT @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups136491112297645522BT_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8500_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > rat] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8501_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A2: set_VEBT_VEBT,F: vEBT_VEBT > code_integer] :
      ( ( member_VEBT_VEBT @ I @ A2 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ A2 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( finite5795047828879050333T_VEBT @ A2 )
         => ( ord_le3102999989581377725nteger @ ( F @ I ) @ ( groups5748017345553531991nteger @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8502_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > code_integer] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_le3102999989581377725nteger @ ( F @ I ) @ ( groups6621422865394947399nteger @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8503_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > rat] :
      ( ( member_int @ I @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8504_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > code_integer] :
      ( ( member_int @ I @ A2 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_le3102999989581377725nteger @ ( F @ I ) @ ( groups7873554091576472773nteger @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8505_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > rat] :
      ( ( member_real @ I @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8506_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > code_integer] :
      ( ( member_real @ I @ A2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_le3102999989581377725nteger @ ( F @ I ) @ ( groups7713935264441627589nteger @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8507_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: nat > rat] :
      ( ( member_nat @ I @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_nat @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8508_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: nat > code_integer] :
      ( ( member_nat @ I @ A2 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X3 ) ) )
       => ( ( finite_finite_nat @ A2 )
         => ( ord_le3102999989581377725nteger @ ( F @ I ) @ ( groups7501900531339628137nteger @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8509_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G @ M )
          @ ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8510_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G @ M )
          @ ( groups3539618377306564664at_int
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8511_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G @ M )
          @ ( groups3542108847815614940at_nat
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8512_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G @ M )
          @ ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8513_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I2: nat] : ( minus_minus_rat @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
          @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff'
thf(fact_8514_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int
          @ ^ [I2: nat] : ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
          @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff'
thf(fact_8515_sum__Suc__diff_H,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( minus_minus_real @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
          @ ( set_or4665077453230672383an_nat @ M @ N ) )
        = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff'
thf(fact_8516_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I2: nat] : ( minus_minus_rat @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_8517_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I2: nat] : ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_8518_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( minus_minus_real @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_8519_sum_OatLeastLessThan__rev,axiom,
    ! [G: nat > nat,N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ ( suc @ I2 ) ) )
        @ ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).

% sum.atLeastLessThan_rev
thf(fact_8520_sum_OatLeastLessThan__rev,axiom,
    ! [G: nat > real,N: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ N @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ ( suc @ I2 ) ) )
        @ ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).

% sum.atLeastLessThan_rev
thf(fact_8521_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I2: nat] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8522_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
     => ( ( ( groups7713935264441627589nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7713935264441627589nteger
                  @ ^ [I2: real] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8523_convex__sum__bound__le,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > code_integer,A: vEBT_VEBT > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
     => ( ( ( groups5748017345553531991nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups5748017345553531991nteger
                  @ ^ [I2: vEBT_VEBT] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8524_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7873554091576472773nteger
                  @ ^ [I2: int] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8525_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
     => ( ( ( groups6621422865394947399nteger @ X @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups6621422865394947399nteger
                  @ ^ [I2: complex] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8526_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I2: nat] : ( times_times_rat @ ( A @ I2 ) @ ( X @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8527_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
     => ( ( ( groups1300246762558778688al_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups1300246762558778688al_rat
                  @ ^ [I2: real] : ( times_times_rat @ ( A @ I2 ) @ ( X @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8528_convex__sum__bound__le,axiom,
    ! [I5: set_VEBT_VEBT,X: vEBT_VEBT > rat,A: vEBT_VEBT > rat,B: rat,Delta: rat] :
      ( ! [I3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
     => ( ( ( groups136491112297645522BT_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups136491112297645522BT_rat
                  @ ^ [I2: vEBT_VEBT] : ( times_times_rat @ ( A @ I2 ) @ ( X @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8529_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
     => ( ( ( groups3906332499630173760nt_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups3906332499630173760nt_rat
                  @ ^ [I2: int] : ( times_times_rat @ ( A @ I2 ) @ ( X @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8530_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: complex > rat,A: complex > rat,B: rat,Delta: rat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
     => ( ( ( groups5058264527183730370ex_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups5058264527183730370ex_rat
                  @ ^ [I2: complex] : ( times_times_rat @ ( A @ I2 ) @ ( X @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8531_sum_Onested__swap,axiom,
    ! [A: nat > nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( groups3542108847815614940at_nat @ ( A @ I2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I2: nat] : ( A @ I2 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% sum.nested_swap
thf(fact_8532_sum_Onested__swap,axiom,
    ! [A: nat > nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( groups6591440286371151544t_real @ ( A @ I2 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( groups6591440286371151544t_real
        @ ^ [J3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( A @ I2 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% sum.nested_swap
thf(fact_8533_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_8534_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_8535_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_8536_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_8537_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_8538_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_8539_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_8540_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > rat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8541_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > int,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8542_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > nat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8543_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > real,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8544_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.head_if
thf(fact_8545_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.head_if
thf(fact_8546_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.head_if
thf(fact_8547_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( plus_p5714425477246183910nteger @ ( groups7501900531339628137nteger @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.head_if
thf(fact_8548_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.head_if
thf(fact_8549_sum_Ohead__if,axiom,
    ! [N: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).

% sum.head_if
thf(fact_8550_sum__le__suminf,axiom,
    ! [F: nat > int,I5: set_nat] :
      ( ( summable_int @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N2: nat] :
              ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8551_sum__le__suminf,axiom,
    ! [F: nat > nat,I5: set_nat] :
      ( ( summable_nat @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N2: nat] :
              ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) ) )
         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8552_sum__le__suminf,axiom,
    ! [F: nat > real,I5: set_nat] :
      ( ( summable_real @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N2: nat] :
              ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) ) )
         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8553_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_8554_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: nat > nat,N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_8555_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: nat > real,N: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ N @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_8556_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_8557_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_8558_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_8559_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_8560_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_8561_summable__Cauchy,axiom,
    ( summable_complex
    = ( ^ [F6: nat > complex] :
        ! [E3: real] :
          ( ( ord_less_real @ zero_zero_real @ E3 )
         => ? [N8: nat] :
            ! [M5: nat] :
              ( ( ord_less_eq_nat @ N8 @ M5 )
             => ! [N4: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F6 @ ( set_or4665077453230672383an_nat @ M5 @ N4 ) ) ) @ E3 ) ) ) ) ) ).

% summable_Cauchy
thf(fact_8562_summable__Cauchy,axiom,
    ( summable_real
    = ( ^ [F6: nat > real] :
        ! [E3: real] :
          ( ( ord_less_real @ zero_zero_real @ E3 )
         => ? [N8: nat] :
            ! [M5: nat] :
              ( ( ord_less_eq_nat @ N8 @ M5 )
             => ! [N4: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F6 @ ( set_or4665077453230672383an_nat @ M5 @ N4 ) ) ) @ E3 ) ) ) ) ) ).

% summable_Cauchy
thf(fact_8563_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > complex] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups2073611262835488442omplex
            @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups2073611262835488442omplex
            @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
thf(fact_8564_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > code_integer] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups7501900531339628137nteger
            @ ^ [K3: nat] : ( minus_8373710615458151222nteger @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_8373710615458151222nteger @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups7501900531339628137nteger
            @ ^ [K3: nat] : ( minus_8373710615458151222nteger @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_z3403309356797280102nteger ) ) ) ).

% sum_natinterval_diff
thf(fact_8565_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_8566_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_8567_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_8568_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_8569_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int
          @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_8570_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real
          @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_8571_summable__partial__sum__bound,axiom,
    ! [F: nat > complex,E: real] :
      ( ( summable_complex @ F )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ~ ! [N9: nat] :
              ~ ! [M2: nat] :
                  ( ( ord_less_eq_nat @ N9 @ M2 )
                 => ! [N10: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M2 @ N10 ) ) ) @ E ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_8572_summable__partial__sum__bound,axiom,
    ! [F: nat > real,E: real] :
      ( ( summable_real @ F )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ~ ! [N9: nat] :
              ~ ! [M2: nat] :
                  ( ( ord_less_eq_nat @ N9 @ M2 )
                 => ! [N10: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M2 @ N10 ) ) ) @ E ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_8573_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_8574_mask__eq__sum__exp,axiom,
    ! [N: nat] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer )
      = ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_8575_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_8576_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_8577_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8578_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8579_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8580_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8581_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8582_sum_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I2: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_8583_sum_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I2: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_8584_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_8585_sum_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_8586_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_8587_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_8588_gauss__sum__nat,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( 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_8589_sum__power2,axiom,
    ! [K: nat] :
      ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).

% sum_power2
thf(fact_8590_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_8591_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_8592_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_8593_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_8594_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_8595_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_8596_double__arith__series,axiom,
    ! [A: rat,D2: rat,N: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I2: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I2 ) @ D2 ) )
          @ ( 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 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_8597_double__arith__series,axiom,
    ! [A: int,D2: int,N: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I2: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I2 ) @ D2 ) )
          @ ( 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 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_8598_double__arith__series,axiom,
    ! [A: complex,D2: complex,N: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I2: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I2 ) @ D2 ) )
          @ ( 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 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_8599_double__arith__series,axiom,
    ! [A: nat,D2: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ D2 ) )
          @ ( 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 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_8600_double__arith__series,axiom,
    ! [A: real,D2: real,N: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I2 ) @ D2 ) )
          @ ( 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 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_8601_arith__series__nat,axiom,
    ! [A: nat,D2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I2 @ D2 ) )
        @ ( 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 @ D2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_8602_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or1269000886237332187st_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_8603_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_8604_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_8605_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_8606_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_8607_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_8608_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_8609_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_8610_arith__series,axiom,
    ! [A: int,D2: int,N: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I2: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I2 ) @ D2 ) )
        @ ( 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 ) @ D2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_8611_arith__series,axiom,
    ! [A: nat,D2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ D2 ) )
        @ ( 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 ) @ D2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_8612_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8613_sum__gp__offset,axiom,
    ! [X: complex,M: nat,N: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8614_sum__gp__offset,axiom,
    ! [X: real,M: nat,N: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8615_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_8616_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_8617_Chebyshev__sum__upper,axiom,
    ! [N: nat,A: nat > rat,B: nat > rat] :
      ( ! [I3: nat,J: nat] :
          ( ( ord_less_eq_nat @ I3 @ J )
         => ( ( ord_less_nat @ J @ N )
           => ( ord_less_eq_rat @ ( A @ I3 ) @ ( A @ J ) ) ) )
     => ( ! [I3: nat,J: nat] :
            ( ( ord_less_eq_nat @ I3 @ J )
           => ( ( ord_less_nat @ J @ N )
             => ( ord_less_eq_rat @ ( B @ J ) @ ( B @ I3 ) ) ) )
       => ( ord_less_eq_rat
          @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N )
            @ ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( times_times_rat @ ( A @ K3 ) @ ( B @ K3 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_rat @ ( groups2906978787729119204at_rat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups2906978787729119204at_rat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper
thf(fact_8618_Chebyshev__sum__upper,axiom,
    ! [N: nat,A: nat > int,B: nat > int] :
      ( ! [I3: nat,J: nat] :
          ( ( ord_less_eq_nat @ I3 @ J )
         => ( ( ord_less_nat @ J @ N )
           => ( ord_less_eq_int @ ( A @ I3 ) @ ( A @ J ) ) ) )
     => ( ! [I3: nat,J: nat] :
            ( ( ord_less_eq_nat @ I3 @ J )
           => ( ( ord_less_nat @ J @ N )
             => ( ord_less_eq_int @ ( B @ J ) @ ( B @ I3 ) ) ) )
       => ( ord_less_eq_int
          @ ( times_times_int @ ( semiri1314217659103216013at_int @ N )
            @ ( groups3539618377306564664at_int
              @ ^ [K3: nat] : ( times_times_int @ ( A @ K3 ) @ ( B @ K3 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_int @ ( groups3539618377306564664at_int @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3539618377306564664at_int @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper
thf(fact_8619_Chebyshev__sum__upper,axiom,
    ! [N: nat,A: nat > real,B: nat > real] :
      ( ! [I3: nat,J: nat] :
          ( ( ord_less_eq_nat @ I3 @ J )
         => ( ( ord_less_nat @ J @ N )
           => ( ord_less_eq_real @ ( A @ I3 ) @ ( A @ J ) ) ) )
     => ( ! [I3: nat,J: nat] :
            ( ( ord_less_eq_nat @ I3 @ J )
           => ( ( ord_less_nat @ J @ N )
             => ( ord_less_eq_real @ ( B @ J ) @ ( B @ I3 ) ) ) )
       => ( ord_less_eq_real
          @ ( times_times_real @ ( semiri5074537144036343181t_real @ N )
            @ ( groups6591440286371151544t_real
              @ ^ [K3: nat] : ( times_times_real @ ( A @ K3 ) @ ( B @ K3 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_real @ ( groups6591440286371151544t_real @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups6591440286371151544t_real @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper
thf(fact_8620_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: nat > nat,B: nat > nat] :
      ( ! [I3: nat,J: nat] :
          ( ( ord_less_eq_nat @ I3 @ J )
         => ( ( ord_less_nat @ J @ N )
           => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J ) ) ) )
     => ( ! [I3: nat,J: nat] :
            ( ( ord_less_eq_nat @ I3 @ J )
           => ( ( ord_less_nat @ J @ N )
             => ( ord_less_eq_nat @ ( B @ J ) @ ( B @ I3 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N
            @ ( groups3542108847815614940at_nat
              @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( B @ I2 ) )
              @ ( 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_8621_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
            @ ^ [P3: 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 ) ) @ P3 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_8622_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
            @ ^ [P3: 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 ) ) @ P3 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_8623_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
            @ ^ [P3: 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 ) ) @ P3 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_8624_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_8625_real__sqrt__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ X )
        = ( sqrt @ Y ) )
      = ( X = Y ) ) ).

% real_sqrt_eq_iff
thf(fact_8626_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I @ K ) ) ).

% lessThan_iff
thf(fact_8627_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I @ K ) ) ).

% lessThan_iff
thf(fact_8628_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_8629_lessThan__iff,axiom,
    ! [I: code_integer,K: code_integer] :
      ( ( member_Code_integer @ I @ ( set_or5754767410780653050nteger @ K ) )
      = ( ord_le6747313008572928689nteger @ I @ K ) ) ).

% lessThan_iff
thf(fact_8630_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_8631_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I @ K ) ) ).

% lessThan_iff
thf(fact_8632_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_8633_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( ( sqrt @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
thf(fact_8634_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% real_sqrt_less_iff
thf(fact_8635_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% real_sqrt_le_iff
thf(fact_8636_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( sqrt @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% real_sqrt_eq_1_iff
thf(fact_8637_real__sqrt__one,axiom,
    ( ( sqrt @ one_one_real )
    = one_one_real ) ).

% real_sqrt_one
thf(fact_8638_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_8639_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_gt_0_iff
thf(fact_8640_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_ge_0_iff
thf(fact_8641_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_8642_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_8643_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_8644_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_8645_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_8646_lessThan__minus__lessThan,axiom,
    ! [N: real,M: real] :
      ( ( minus_minus_set_real @ ( set_or5984915006950818249n_real @ N ) @ ( set_or5984915006950818249n_real @ M ) )
      = ( set_or66887138388493659n_real @ M @ N ) ) ).

% lessThan_minus_lessThan
thf(fact_8647_lessThan__minus__lessThan,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( set_ord_lessThan_nat @ M ) )
      = ( set_or4665077453230672383an_nat @ M @ N ) ) ).

% lessThan_minus_lessThan
thf(fact_8648_lessThan__minus__lessThan,axiom,
    ! [N: int,M: int] :
      ( ( minus_minus_set_int @ ( set_ord_lessThan_int @ N ) @ ( set_ord_lessThan_int @ M ) )
      = ( set_or4662586982721622107an_int @ M @ N ) ) ).

% lessThan_minus_lessThan
thf(fact_8649_lessThan__minus__lessThan,axiom,
    ! [N: code_integer,M: code_integer] :
      ( ( minus_2355218937544613996nteger @ ( set_or5754767410780653050nteger @ N ) @ ( set_or5754767410780653050nteger @ M ) )
      = ( set_or8404916559141939852nteger @ M @ N ) ) ).

% lessThan_minus_lessThan
thf(fact_8650_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_8651_real__sqrt__mult__self,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
      = ( abs_abs_real @ A ) ) ).

% real_sqrt_mult_self
thf(fact_8652_real__sqrt__abs2,axiom,
    ! [X: real] :
      ( ( sqrt @ ( times_times_real @ X @ X ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs2
thf(fact_8653_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_8654_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_8655_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_8656_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_8657_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_8658_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_8659_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_8660_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_8661_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa3: 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 @ Xa3 @ ( 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 @ Xa3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_8662_Complex__sum_H,axiom,
    ! [F: nat > real,S: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [X2: nat] : ( complex2 @ ( F @ X2 ) @ zero_zero_real )
        @ S )
      = ( complex2 @ ( groups6591440286371151544t_real @ F @ S ) @ zero_zero_real ) ) ).

% Complex_sum'
thf(fact_8663_Complex__sum_H,axiom,
    ! [F: complex > real,S: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X2: complex] : ( complex2 @ ( F @ X2 ) @ zero_zero_real )
        @ S )
      = ( complex2 @ ( groups5808333547571424918x_real @ F @ S ) @ zero_zero_real ) ) ).

% Complex_sum'
thf(fact_8664_real__sqrt__minus,axiom,
    ! [X: real] :
      ( ( sqrt @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_minus
thf(fact_8665_int__sum,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X2: int] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% int_sum
thf(fact_8666_int__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
        @ A2 ) ) ).

% int_sum
thf(fact_8667_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_8668_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_less_mono
thf(fact_8669_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_le_mono
thf(fact_8670_real__sqrt__divide,axiom,
    ! [X: real,Y: real] :
      ( ( sqrt @ ( divide_divide_real @ X @ Y ) )
      = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_divide
thf(fact_8671_real__sqrt__power,axiom,
    ! [X: real,K: nat] :
      ( ( sqrt @ ( power_power_real @ X @ K ) )
      = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).

% real_sqrt_power
thf(fact_8672_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X2: rat] : ( ord_less_rat @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8673_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X2: num] : ( ord_less_num @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8674_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X2: int] : ( ord_less_int @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8675_lessThan__def,axiom,
    ( set_or5754767410780653050nteger
    = ( ^ [U2: code_integer] :
          ( collect_Code_integer
          @ ^ [X2: code_integer] : ( ord_le6747313008572928689nteger @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8676_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8677_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X2: real] : ( ord_less_real @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8678_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_gt_zero
thf(fact_8679_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sqrt @ X )
          = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_8680_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_zero
thf(fact_8681_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_8682_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_8683_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
      = ( ord_less_rat @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8684_lessThan__strict__subset__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8685_lessThan__strict__subset__iff,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
      = ( ord_less_int @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8686_lessThan__strict__subset__iff,axiom,
    ! [M: code_integer,N: code_integer] :
      ( ( ord_le1307284697595431911nteger @ ( set_or5754767410780653050nteger @ M ) @ ( set_or5754767410780653050nteger @ N ) )
      = ( ord_le6747313008572928689nteger @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8687_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8688_lessThan__strict__subset__iff,axiom,
    ! [M: real,N: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
      = ( ord_less_real @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8689_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_8690_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_8691_sum__SucD,axiom,
    ! [F: nat > nat,A2: set_nat,N: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A2 )
        = ( suc @ N ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A2 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).

% sum_SucD
thf(fact_8692_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y5: int] :
                  ( ( member_int @ Y5 @ A2 )
                 => ( ( X2 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8693_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y5: complex] :
                  ( ( member_complex @ Y5 @ A2 )
                 => ( ( X2 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8694_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
              & ( ( F @ X2 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y5: nat] :
                  ( ( member_nat @ Y5 @ A2 )
                 => ( ( X2 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8695_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y5: int] :
                  ( ( member_int @ Y5 @ A2 )
                 => ( ( X2 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8696_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y5: complex] :
                  ( ( member_complex @ Y5 @ A2 )
                 => ( ( X2 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8697_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
              & ( ( F @ X2 )
                = one_one_nat )
              & ! [Y5: nat] :
                  ( ( member_nat @ Y5 @ A2 )
                 => ( ( X2 != Y5 )
                   => ( ( F @ Y5 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8698_real__div__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
        = ( sqrt @ X ) ) ) ).

% real_div_sqrt
thf(fact_8699_sqrt__add__le__add__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 @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_8700_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_8701_lessThan__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).

% lessThan_nat_numeral
thf(fact_8702_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_8703_sum_Onat__diff__reindex,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_8704_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_8705_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
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8706_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
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8707_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
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8708_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
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8709_sum__lessThan__telescope_H,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N4: nat] : ( minus_minus_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8710_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N4: nat] : ( minus_minus_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8711_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8712_sum__lessThan__telescope,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N4: nat] : ( minus_minus_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8713_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N4: nat] : ( minus_minus_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8714_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N4: nat] : ( minus_minus_real @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8715_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
        @ ^ [I2: nat] : ( minus_minus_rat @ ( F @ I2 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8716_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
        @ ^ [I2: nat] : ( minus_minus_int @ ( F @ I2 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8717_sumr__diff__mult__const2,axiom,
    ! [F: nat > complex,N: nat,R2: complex] :
      ( ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ R2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [I2: nat] : ( minus_minus_complex @ ( F @ I2 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8718_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
        @ ^ [I2: nat] : ( minus_minus_real @ ( F @ I2 ) @ R2 )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8719_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_8720_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_8721_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_8722_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_8723_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_8724_sum_Onat__group,axiom,
    ! [G: nat > nat,K: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [M5: nat] : ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M5 @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M5 @ K ) @ K ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N @ K ) ) ) ) ).

% sum.nat_group
thf(fact_8725_sum_Onat__group,axiom,
    ! [G: nat > real,K: nat,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M5: nat] : ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M5 @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M5 @ K ) @ K ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N @ K ) ) ) ) ).

% sum.nat_group
thf(fact_8726_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X2: complex] : X2
          @ ( collect_complex
            @ ^ [Z7: complex] :
                ( ( power_power_complex @ Z7 @ N )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_8727_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_8728_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_8729_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_8730_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_8731_power__diff__1__eq,axiom,
    ! [X: code_integer,N: nat] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ N ) @ one_one_Code_integer )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_8732_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_8733_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_8734_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_8735_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_8736_one__diff__power__eq,axiom,
    ! [X: code_integer,N: nat] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_8737_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_8738_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_8739_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_8740_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_8741_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_8742_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_8743_suminf__split__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real @ F )
        = ( plus_plus_real
          @ ( suminf_real
            @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K ) ) )
          @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_split_initial_segment
thf(fact_8744_suminf__minus__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_minus_initial_segment
thf(fact_8745_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_8746_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_8747_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_8748_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_8749_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_8750_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D2: 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 @ D2 ) @ ( 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 @ D2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_8751_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_8752_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_8753_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_8754_sum__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X2: complex] : X2
          @ ( collect_complex
            @ ^ [Z7: complex] :
                ( ( power_power_complex @ Z7 @ N )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_8755_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_8756_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_8757_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_8758_sum__less__suminf,axiom,
    ! [F: nat > int,N: nat] :
      ( ( summable_int @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N @ M3 )
           => ( ord_less_int @ zero_zero_int @ ( F @ M3 ) ) )
       => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_8759_sum__less__suminf,axiom,
    ! [F: nat > nat,N: nat] :
      ( ( summable_nat @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N @ M3 )
           => ( ord_less_nat @ zero_zero_nat @ ( F @ M3 ) ) )
       => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_8760_sum__less__suminf,axiom,
    ! [F: nat > real,N: nat] :
      ( ( summable_real @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N @ M3 )
           => ( ord_less_real @ zero_zero_real @ ( F @ M3 ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_8761_lemma__termdiff1,axiom,
    ! [Z: complex,H2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P3: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_power_complex @ Z @ P3 ) ) @ ( power_power_complex @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [P3: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P3 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P3 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8762_lemma__termdiff1,axiom,
    ! [Z: code_integer,H2: code_integer,M: nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [P3: nat] : ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ Z @ H2 ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_8256067586552552935nteger @ Z @ P3 ) ) @ ( power_8256067586552552935nteger @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups7501900531339628137nteger
        @ ^ [P3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Z @ P3 ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ Z @ H2 ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_8256067586552552935nteger @ Z @ ( minus_minus_nat @ M @ P3 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8763_lemma__termdiff1,axiom,
    ! [Z: rat,H2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [P3: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_power_rat @ Z @ P3 ) ) @ ( power_power_rat @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [P3: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P3 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P3 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8764_lemma__termdiff1,axiom,
    ! [Z: int,H2: int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P3: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_power_int @ Z @ P3 ) ) @ ( power_power_int @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups3539618377306564664at_int
        @ ^ [P3: nat] : ( times_times_int @ ( power_power_int @ Z @ P3 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P3 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8765_lemma__termdiff1,axiom,
    ! [Z: real,H2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P3: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_power_real @ Z @ P3 ) ) @ ( power_power_real @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [P3: nat] : ( times_times_real @ ( power_power_real @ Z @ P3 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P3 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P3 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8766_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_8767_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_8768_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_8769_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
          @ ^ [P3: nat] : ( times_times_complex @ ( power_power_complex @ X @ P3 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ P3 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8770_diff__power__eq__sum,axiom,
    ! [X: code_integer,N: nat,Y: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ ( suc @ N ) ) @ ( power_8256067586552552935nteger @ Y @ ( suc @ N ) ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ Y )
        @ ( groups7501900531339628137nteger
          @ ^ [P3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ P3 ) @ ( power_8256067586552552935nteger @ Y @ ( minus_minus_nat @ N @ P3 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8771_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
          @ ^ [P3: nat] : ( times_times_rat @ ( power_power_rat @ X @ P3 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ P3 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8772_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
          @ ^ [P3: nat] : ( times_times_int @ ( power_power_int @ X @ P3 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ P3 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8773_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
          @ ^ [P3: nat] : ( times_times_real @ ( power_power_real @ X @ P3 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ P3 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8774_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
          @ ^ [I2: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) ) @ ( power_power_complex @ X @ I2 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8775_power__diff__sumr2,axiom,
    ! [X: code_integer,N: nat,Y: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ Y )
        @ ( groups7501900531339628137nteger
          @ ^ [I2: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Y @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) ) @ ( power_8256067586552552935nteger @ X @ I2 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8776_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
          @ ^ [I2: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) ) @ ( power_power_rat @ X @ I2 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8777_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
          @ ^ [I2: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) ) @ ( power_power_int @ X @ I2 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8778_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
          @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) ) @ ( power_power_real @ X @ I2 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8779_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_8780_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_8781_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_8782_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_8783_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_8784_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_8785_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_8786_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_8787_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > code_integer,K6: code_integer,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_le3102999989581377725nteger @ ( F @ P7 ) @ K6 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ K6 )
       => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8788_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > rat,K6: rat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_rat @ ( F @ P7 ) @ K6 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ K6 )
       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8789_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > int,K6: int,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_int @ ( F @ P7 ) @ K6 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ K6 )
       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8790_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > nat,K6: nat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_nat @ ( F @ P7 ) @ K6 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ K6 )
       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8791_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > real,K6: real,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_real @ ( F @ P7 ) @ K6 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ K6 )
       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K6 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8792_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_8793_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_8794_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_8795_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_8796_sum__less__suminf2,axiom,
    ! [F: nat > int,N: nat,I: nat] :
      ( ( summable_int @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N @ M3 )
           => ( ord_less_eq_int @ zero_zero_int @ ( F @ M3 ) ) )
       => ( ( 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_8797_sum__less__suminf2,axiom,
    ! [F: nat > nat,N: nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N @ M3 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M3 ) ) )
       => ( ( 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_8798_sum__less__suminf2,axiom,
    ! [F: nat > real,N: nat,I: nat] :
      ( ( summable_real @ F )
     => ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ N @ M3 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ M3 ) ) )
       => ( ( 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_8799_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
          @ ^ [I2: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8800_one__diff__power__eq_H,axiom,
    ! [X: code_integer,N: nat] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X )
        @ ( groups7501900531339628137nteger
          @ ^ [I2: nat] : ( power_8256067586552552935nteger @ X @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8801_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
          @ ^ [I2: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8802_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
          @ ^ [I2: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8803_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
          @ ^ [I2: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8804_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_8805_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_8806_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa3: 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 @ Xa3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_8807_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_8808_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_8809_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_8810_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ ( F @ I2 ) @ ( G @ I2 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_split_even_odd
thf(fact_8811_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_8812_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_8813_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_8814_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_8815_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_8816_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_8817_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_8818_arctan__half,axiom,
    ( arctan
    = ( ^ [X2: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X2 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_8819_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X2: int] : X2
          @ ( set_or1266510415728281911st_int @ M @ N ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_8820_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( 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 @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_8821_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_8822_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_8823_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_8824_sumr__cos__zero__one,axiom,
    ! [N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ zero_zero_real @ M5 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_8825_arcosh__def,axiom,
    ( arcosh_real
    = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arcosh_def
thf(fact_8826_freeze__rule,axiom,
    ! [A: array_VEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( hoare_3904069481286416050_VEBTi @ ( snga_assn_VEBT_VEBTi @ A @ Xs ) @ ( array_8141364883501958055_VEBTi @ A )
      @ ^ [R5: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ A @ Xs ) @ ( pure_assn @ ( R5 = Xs ) ) ) ) ).

% freeze_rule
thf(fact_8827_of__real__0,axiom,
    ( ( real_V1803761363581548252l_real @ zero_zero_real )
    = zero_zero_real ) ).

% of_real_0
thf(fact_8828_of__real__0,axiom,
    ( ( real_V4546457046886955230omplex @ zero_zero_real )
    = zero_zero_complex ) ).

% of_real_0
thf(fact_8829_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( ( real_V1803761363581548252l_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
thf(fact_8830_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( ( real_V4546457046886955230omplex @ X )
        = zero_zero_complex )
      = ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
thf(fact_8831_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% of_real_numeral
thf(fact_8832_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
      = ( numera6690914467698888265omplex @ W ) ) ).

% of_real_numeral
thf(fact_8833_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_8834_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_8835_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_8836_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_8837_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_8838_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_8839_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_8840_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_8841_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_8842_sin__of__real__pi,axiom,
    ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
    = zero_zero_real ) ).

% sin_of_real_pi
thf(fact_8843_sin__of__real__pi,axiom,
    ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
    = zero_zero_complex ) ).

% sin_of_real_pi
thf(fact_8844_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_8845_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_8846_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_8847_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_8848_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_8849_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_8850_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_8851_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_8852_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_8853_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_8854_complex__of__real__def,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [R5: real] : ( complex2 @ R5 @ zero_zero_real ) ) ) ).

% complex_of_real_def
thf(fact_8855_complex__of__real__code,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [X2: real] : ( complex2 @ X2 @ zero_zero_real ) ) ) ).

% complex_of_real_code
thf(fact_8856_complex__eq__cancel__iff2,axiom,
    ! [X: real,Y: real,Xa3: real] :
      ( ( ( complex2 @ X @ Y )
        = ( real_V4546457046886955230omplex @ Xa3 ) )
      = ( ( X = Xa3 )
        & ( Y = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
thf(fact_8857_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_8858_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_8859_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_8860_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_8861_complex__of__real__add__Complex,axiom,
    ! [R2: real,X: real,Y: real] :
      ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X @ Y ) )
      = ( complex2 @ ( plus_plus_real @ R2 @ X ) @ Y ) ) ).

% complex_of_real_add_Complex
thf(fact_8862_Complex__add__complex__of__real,axiom,
    ! [X: real,Y: real,R2: real] :
      ( ( plus_plus_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ ( plus_plus_real @ X @ R2 ) @ Y ) ) ).

% Complex_add_complex_of_real
thf(fact_8863_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_8864_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_8865_cos__int__times__real,axiom,
    ! [M: int,X: real] :
      ( ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
      = ( real_V1803761363581548252l_real @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).

% cos_int_times_real
thf(fact_8866_cos__int__times__real,axiom,
    ! [M: int,X: real] :
      ( ( cos_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
      = ( real_V4546457046886955230omplex @ ( cos_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).

% cos_int_times_real
thf(fact_8867_sin__int__times__real,axiom,
    ! [M: int,X: real] :
      ( ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ ( real_V1803761363581548252l_real @ X ) ) )
      = ( real_V1803761363581548252l_real @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).

% sin_int_times_real
thf(fact_8868_sin__int__times__real,axiom,
    ! [M: int,X: real] :
      ( ( sin_complex @ ( times_times_complex @ ( ring_17405671764205052669omplex @ M ) @ ( real_V4546457046886955230omplex @ X ) ) )
      = ( real_V4546457046886955230omplex @ ( sin_real @ ( times_times_real @ ( ring_1_of_int_real @ M ) @ X ) ) ) ) ).

% sin_int_times_real
thf(fact_8869_cos__sin__eq,axiom,
    ( cos_real
    = ( ^ [X2: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ).

% cos_sin_eq
thf(fact_8870_cos__sin__eq,axiom,
    ( cos_complex
    = ( ^ [X2: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ).

% cos_sin_eq
thf(fact_8871_sin__cos__eq,axiom,
    ( sin_real
    = ( ^ [X2: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ).

% sin_cos_eq
thf(fact_8872_sin__cos__eq,axiom,
    ( sin_complex
    = ( ^ [X2: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ).

% sin_cos_eq
thf(fact_8873_minus__sin__cos__eq,axiom,
    ! [X: real] :
      ( ( uminus_uminus_real @ ( sin_real @ X ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_8874_minus__sin__cos__eq,axiom,
    ! [X: complex] :
      ( ( uminus1482373934393186551omplex @ ( sin_complex @ X ) )
      = ( cos_complex @ ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_8875_arsinh__def,axiom,
    ( arsinh_real
    = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arsinh_def
thf(fact_8876_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T4: real] :
            ( ( ord_less_real @ zero_zero_real @ T4 )
            & ( ord_less_real @ T4 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( 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_8877_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ? [T4: real] :
            ( ( ord_less_real @ X @ T4 )
            & ( ord_less_real @ T4 @ zero_zero_real )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( 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_8878_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
      & ( ( cos_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( 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_8879_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_8880_arcsin__0,axiom,
    ( ( arcsin @ zero_zero_real )
    = zero_zero_real ) ).

% arcsin_0
thf(fact_8881_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_8882_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_8883_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_8884_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_8885_fact__0,axiom,
    ( ( semiri5044797733671781792omplex @ zero_zero_nat )
    = one_one_complex ) ).

% fact_0
thf(fact_8886_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_8887_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_8888_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_8889_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_8890_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_8891_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_Suc
thf(fact_8892_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_Suc
thf(fact_8893_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_Suc
thf(fact_8894_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_Suc
thf(fact_8895_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri5044797733671781792omplex @ ( suc @ N ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).

% fact_Suc
thf(fact_8896_fact__2,axiom,
    ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8897_fact__2,axiom,
    ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8898_fact__2,axiom,
    ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8899_fact__2,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8900_fact__2,axiom,
    ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8901_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_8902_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_8903_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ N )
     != zero_zero_rat ) ).

% fact_nonzero
thf(fact_8904_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ N )
     != zero_zero_int ) ).

% fact_nonzero
thf(fact_8905_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri3624122377584611663nteger @ N )
     != zero_z3403309356797280102nteger ) ).

% fact_nonzero
thf(fact_8906_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ N )
     != zero_zero_real ) ).

% fact_nonzero
thf(fact_8907_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ N )
     != zero_zero_nat ) ).

% fact_nonzero
thf(fact_8908_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri5044797733671781792omplex @ N )
     != zero_zero_complex ) ).

% fact_nonzero
thf(fact_8909_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_zero
thf(fact_8910_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri3624122377584611663nteger @ N ) ) ).

% fact_ge_zero
thf(fact_8911_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_zero
thf(fact_8912_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_zero
thf(fact_8913_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_zero
thf(fact_8914_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_8915_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_8916_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( semiri3624122377584611663nteger @ N ) @ zero_z3403309356797280102nteger ) ).

% fact_not_neg
thf(fact_8917_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_8918_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_8919_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_gt_zero
thf(fact_8920_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_gt_zero
thf(fact_8921_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri3624122377584611663nteger @ N ) ) ).

% fact_gt_zero
thf(fact_8922_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_gt_zero
thf(fact_8923_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_gt_zero
thf(fact_8924_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).

% fact_less_mono
thf(fact_8925_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).

% fact_less_mono
thf(fact_8926_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_le6747313008572928689nteger @ ( semiri3624122377584611663nteger @ M ) @ ( semiri3624122377584611663nteger @ N ) ) ) ) ).

% fact_less_mono
thf(fact_8927_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).

% fact_less_mono
thf(fact_8928_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono
thf(fact_8929_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
        = zero_zero_int ) ) ).

% fact_mod
thf(fact_8930_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
        = zero_z3403309356797280102nteger ) ) ).

% fact_mod
thf(fact_8931_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
        = zero_zero_nat ) ) ).

% fact_mod
thf(fact_8932_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8933_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8934_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8935_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8936_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_8937_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_8938_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_8939_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% choose_dvd
thf(fact_8940_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% choose_dvd
thf(fact_8941_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% choose_dvd
thf(fact_8942_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% choose_dvd
thf(fact_8943_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8944_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
      = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8945_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
      = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8946_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8947_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8948_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_8949_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_8950_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_8951_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M5 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8952_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8953_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8954_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M5 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8955_fact__num__eq__if,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M5 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8956_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_8957_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_8958_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_8959_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_8960_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_8961_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri5044797733671781792omplex @ N )
        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8962_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > complex > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).

% Maclaurin_zero
thf(fact_8963_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > real > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).

% Maclaurin_zero
thf(fact_8964_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > rat > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).

% Maclaurin_zero
thf(fact_8965_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > nat > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).

% Maclaurin_zero
thf(fact_8966_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > int > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).

% Maclaurin_zero
thf(fact_8967_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > code_integer > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_z3403309356797280102nteger ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_z3403309356797280102nteger ) ) ) ) ).

% Maclaurin_zero
thf(fact_8968_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J2: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B7: real] :
          ( ( F @ H2 )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( J2 @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
              @ ( set_ord_lessThan_nat @ N ) )
            @ ( times_times_real @ B7 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_8969_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_8970_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_8971_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_8972_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_8973_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_8974_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_8975_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_8976_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_8977_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_8978_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_8979_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ? [T4: real] :
            ( ( ord_less_real @ zero_zero_real @ T4 )
            & ( ord_less_real @ T4 @ X )
            & ( ( sin_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( 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_8980_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [T4: real] :
          ( ( ord_less_real @ zero_zero_real @ T4 )
          & ( ord_less_eq_real @ T4 @ X )
          & ( ( sin_real @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( 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_8981_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
      & ( ( sin_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( 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_8982_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] :
      ( ( sin_real @ X )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( 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_8983_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_8984_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono_nat
thf(fact_8985_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_8986_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_8987_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_8988_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_8989_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_8990_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_8991_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T4: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T4 ) )
            & ( ord_less_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
            & ( ( exp_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_8992_sin__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
      @ ( sin_real @ X ) ) ).

% sin_paired
thf(fact_8993_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_8994_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_8995_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).

% exp_less_mono
thf(fact_8996_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% exp_less_cancel_iff
thf(fact_8997_sums__zero,axiom,
    ( sums_complex
    @ ^ [N4: nat] : zero_zero_complex
    @ zero_zero_complex ) ).

% sums_zero
thf(fact_8998_sums__zero,axiom,
    ( sums_real
    @ ^ [N4: nat] : zero_zero_real
    @ zero_zero_real ) ).

% sums_zero
thf(fact_8999_sums__zero,axiom,
    ( sums_nat
    @ ^ [N4: nat] : zero_zero_nat
    @ zero_zero_nat ) ).

% sums_zero
thf(fact_9000_sums__zero,axiom,
    ( sums_int
    @ ^ [N4: nat] : zero_zero_int
    @ zero_zero_int ) ).

% sums_zero
thf(fact_9001_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_9002_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_9003_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ X )
        = one_one_real )
      = ( X = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_9004_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_9005_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_9006_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_9007_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_9008_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_9009_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ ( ln_ln_real @ X ) )
        = X )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% exp_ln_iff
thf(fact_9010_exp__ln,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( exp_real @ ( ln_ln_real @ X ) )
        = X ) ) ).

% exp_ln
thf(fact_9011_powser__sums__zero__iff,axiom,
    ! [A: nat > complex,X: complex] :
      ( ( sums_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( A @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_9012_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X: real] :
      ( ( sums_real
        @ ^ [N4: nat] : ( times_times_real @ ( A @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_9013_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_9014_sums__single,axiom,
    ! [I: nat,F: nat > int] :
      ( sums_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_9015_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% exp_less_cancel
thf(fact_9016_not__exp__less__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_9017_exp__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_gt_zero
thf(fact_9018_exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( exp_real @ X3 )
          = Y ) ) ).

% exp_total
thf(fact_9019_exp__ge__zero,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_ge_zero
thf(fact_9020_not__exp__le__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_9021_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_9022_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_9023_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_9024_lemma__exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y @ one_one_real ) )
          & ( ( exp_real @ X3 )
            = Y ) ) ) ).

% lemma_exp_total
thf(fact_9025_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).

% ln_ge_iff
thf(fact_9026_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_9027_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_9028_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_9029_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_9030_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_9031_arccos__cos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( arccos @ ( cos_real @ X ) )
          = X ) ) ) ).

% arccos_cos
thf(fact_9032_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_9033_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_9034_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_9035_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_9036_arccos__cos2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( arccos @ ( cos_real @ X ) )
          = ( uminus_uminus_real @ X ) ) ) ) ).

% arccos_cos2
thf(fact_9037_power__half__series,axiom,
    ( sums_real
    @ ^ [N4: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N4 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_9038_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_9039_sums__if_H,axiom,
    ! [G: nat > real,X: real] :
      ( ( sums_real @ G @ X )
     => ( sums_real
        @ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X ) ) ).

% sums_if'
thf(fact_9040_sums__if,axiom,
    ! [G: nat > real,X: real,F: nat > real,Y: real] :
      ( ( sums_real @ G @ X )
     => ( ( sums_real @ F @ Y )
       => ( sums_real
          @ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( F @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sums_if
thf(fact_9041_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_9042_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_9043_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_9044_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_9045_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_9046_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
      & ( ( exp_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_9047_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_9048_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_9049_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_9050_cos__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
      @ ( cos_real @ X ) ) ).

% cos_paired
thf(fact_9051_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_9052_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] :
          ( ( arccos @ ( cos_real @ Theta ) )
         != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).

% arccos_cos_eq_abs_2pi
thf(fact_9053_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A @ B ) )
      = zero_zero_nat ) ).

% VEBT_internal.height.simps(1)
thf(fact_9054_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L3: int] :
          ( if_int
          @ ( ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L3
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K3 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_9055_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% or_nonnegative_int_iff
thf(fact_9056_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        | ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_9057_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_9058_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_9059_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_9060_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).

% or_greater_eq
thf(fact_9061_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_9062_prod__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_fact
thf(fact_9063_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_9064_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
          @ ( groups708209901874060359at_nat
            @ ^ [X2: nat] : X2
            @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_9065_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
        = ( groups708209901874060359at_nat
          @ ^ [X2: nat] : X2
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_9066_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_9067_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L3: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_9068_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_9069_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_9070_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_9071_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_9072_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_9073_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_9074_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_9075_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_9076_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_9077_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_9078_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_9079_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_9080_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_9081_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_9082_prod__int__plus__eq,axiom,
    ! [I: nat,J2: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J2 ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : X2
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J2 ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_9083_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_9084_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_9085_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_9086_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_9087_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_9088_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9089_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
            & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_9090_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_9091_floor__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
     => ( ( ord_less_nat @ K @ ( 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 @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_9092_binomial__code,axiom,
    ( binomial
    = ( ^ [N4: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K3 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_9093_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X )
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( 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_9094_binomial__Suc__n,axiom,
    ! [N: nat] :
      ( ( binomial @ ( suc @ N ) @ N )
      = ( suc @ N ) ) ).

% binomial_Suc_n
thf(fact_9095_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_9096_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_9097_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( binomial @ N @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N @ K ) ) ).

% binomial_eq_0_iff
thf(fact_9098_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_9099_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_9100_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
      = ( ord_less_eq_nat @ K @ N ) ) ).

% zero_less_binomial_iff
thf(fact_9101_real__sqrt__inverse,axiom,
    ! [X: real] :
      ( ( sqrt @ ( inverse_inverse_real @ X ) )
      = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_inverse
thf(fact_9102_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( binomial @ N @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_9103_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).

% Suc_times_binomial
thf(fact_9104_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_9105_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_9106_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_9107_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X2: real,Y5: real] : ( times_times_real @ X2 @ ( inverse_inverse_real @ Y5 ) ) ) ) ).

% divide_real_def
thf(fact_9108_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).

% zero_less_binomial
thf(fact_9109_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_9110_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_9111_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_9112_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_9113_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
        = ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).

% inverse_powr
thf(fact_9114_binomial__absorption,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_9115_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
        = ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% binomial_fact_lemma
thf(fact_9116_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_9117_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_9118_real__arch__inverse,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
      = ( ? [N4: nat] :
            ( ( N4 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ E ) ) ) ) ).

% real_arch_inverse
thf(fact_9119_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
        = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_9120_ln__inverse,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
        = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).

% ln_inverse
thf(fact_9121_binomial__maximum,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_9122_binomial__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K4 @ N )
         => ( ord_less_eq_nat @ ( binomial @ N @ K4 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_9123_binomial__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) @ N )
       => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K4 ) ) ) ) ).

% binomial_mono
thf(fact_9124_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% binomial_maximum'
thf(fact_9125_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% binomial_le_pow2
thf(fact_9126_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_9127_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_9128_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_9129_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_9130_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_9131_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( ( ord_less_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) @ N )
       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K4 ) ) ) ) ).

% binomial_strict_mono
thf(fact_9132_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( ( ord_less_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K4 @ N )
         => ( ord_less_nat @ ( binomial @ N @ K4 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_9133_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_9134_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( binomial @ N @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_9135_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_9136_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_9137_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_9138_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_9139_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_9140_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_9141_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_9142_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_9143_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_9144_divide__complex__def,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X2: complex,Y5: complex] : ( times_times_complex @ X2 @ ( invers8013647133539491842omplex @ Y5 ) ) ) ) ).

% divide_complex_def
thf(fact_9145_real__scaleR__def,axiom,
    real_V1485227260804924795R_real = times_times_real ).

% real_scaleR_def
thf(fact_9146_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_9147_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_9148_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_9149_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_9150_atMost__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).

% atMost_nat_numeral
thf(fact_9151_sum__choose__upper,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ K3 @ M )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_9152_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_9153_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_9154_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_9155_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_9156_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R2 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).

% vandermonde
thf(fact_9157_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_9158_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_9159_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_9160_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A @ I3 )
            = zero_zero_nat ) )
     => ( ! [J: nat] :
            ( ( ord_less_nat @ N @ J )
           => ( ( B @ J )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( power_power_nat @ X @ I2 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( 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 @ M @ N ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_9161_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_9162_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_9163_binomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% binomial_r_part_sum
thf(fact_9164_choose__linear__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( times_times_nat @ I2 @ ( binomial @ N @ I2 ) )
        @ ( 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_9165_sinh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sinh_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% sinh_real_zero_iff
thf(fact_9166_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% sinh_real_less_iff
thf(fact_9167_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N4: nat] : N4 ) ) ).

% of_nat_id
thf(fact_9168_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% sinh_real_neg_iff
thf(fact_9169_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% sinh_real_pos_iff
thf(fact_9170_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_9171_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% sinh_real_nonneg_iff
thf(fact_9172_sinh__less__cosh__real,axiom,
    ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).

% sinh_less_cosh_real
thf(fact_9173_cosh__real__nonzero,axiom,
    ! [X: real] :
      ( ( cosh_real @ X )
     != zero_zero_real ) ).

% cosh_real_nonzero
thf(fact_9174_cosh__real__pos,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_pos
thf(fact_9175_cosh__real__nonpos__le__iff,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 @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_9176_cosh__real__nonneg__le__iff,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 @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_9177_cosh__real__nonneg,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_nonneg
thf(fact_9178_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_9179_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_9180_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_9181_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( arcosh_real @ ( cosh_real @ X ) )
        = X ) ) ).

% arcosh_cosh_real
thf(fact_9182_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_9183_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_9184_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_9185_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_9186_divide__i,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).

% divide_i
thf(fact_9187_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_9188_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_9189_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_9190_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_9191_less__eq__mask,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).

% less_eq_mask
thf(fact_9192_mask__nonnegative__int,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).

% mask_nonnegative_int
thf(fact_9193_not__mask__negative__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_9194_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_9195_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_9196_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_9197_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_9198_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_9199_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_9200_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_9201_Complex__eq,axiom,
    ( complex2
    = ( ^ [A4: real,B2: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A4 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).

% Complex_eq
thf(fact_9202_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_9203_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N4: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_9204_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N4: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_9205_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R: real,A6: real] :
      ( Z
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A6 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A6 ) ) ) ) ) ) ).

% complex_split_polar
thf(fact_9206_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_9207_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_9208_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_9209_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_9210_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_9211_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_9212_Arg__zero,axiom,
    ( ( arg @ zero_zero_complex )
    = zero_zero_real ) ).

% Arg_zero
thf(fact_9213_of__real__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
        = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).

% of_real_sqrt
thf(fact_9214_Arg__bounded,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
      & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).

% Arg_bounded
thf(fact_9215_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_9216_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_9217_cis__zero,axiom,
    ( ( cis @ zero_zero_real )
    = one_one_complex ) ).

% cis_zero
thf(fact_9218_cot__pi,axiom,
    ( ( cot_real @ pi )
    = zero_zero_real ) ).

% cot_pi
thf(fact_9219_cot__npi,axiom,
    ! [N: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_9220_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_9221_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_9222_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_9223_cis__mult,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).

% cis_mult
thf(fact_9224_cis__divide,axiom,
    ! [A: real,B: real] :
      ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).

% cis_divide
thf(fact_9225_DeMoivre,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_complex @ ( cis @ A ) @ N )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).

% DeMoivre
thf(fact_9226_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_9227_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_9228_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
          @ ^ [Z7: complex] :
              ( ( power_power_complex @ Z7 @ N )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9229_divmod__BitM__2__eq,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
      = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).

% divmod_BitM_2_eq
thf(fact_9230_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_9231_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_9232_semiring__norm_I28_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ N ) ) ) ).

% semiring_norm(28)
thf(fact_9233_semiring__norm_I27_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit0 @ N ) )
      = ( bit1 @ ( bitM @ N ) ) ) ).

% semiring_norm(27)
thf(fact_9234_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_9235_BitM__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ ( bitM @ N ) @ one )
      = ( bit0 @ N ) ) ).

% BitM_plus_one
thf(fact_9236_one__plus__BitM,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% one_plus_BitM
thf(fact_9237_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_9238_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_9239_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_9240_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_9241_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(8)
thf(fact_9242_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(4)
thf(fact_9243_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(3)
thf(fact_9244_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(7)
thf(fact_9245_take__bit__minus,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).

% take_bit_minus
thf(fact_9246_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_9247_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_9248_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).

% take_bit_diff
thf(fact_9249_take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L ) ) ) ).

% take_bit_mult
thf(fact_9250_add__inc,axiom,
    ! [X: num,Y: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).

% add_inc
thf(fact_9251_num__induct,axiom,
    ! [P: num > $o,X: num] :
      ( ( P @ one )
     => ( ! [X3: num] :
            ( ( P @ X3 )
           => ( P @ ( inc @ X3 ) ) )
       => ( P @ X ) ) ) ).

% num_induct
thf(fact_9252_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_9253_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_9254_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_9255_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).

% take_bit_nonnegative
thf(fact_9256_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_9257_not__take__bit__negative,axiom,
    ! [N: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_9258_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_9259_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_9260_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
      = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(6)
thf(fact_9261_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
      = ( bit1 @ M ) ) ).

% or_not_num_neg.simps(3)
thf(fact_9262_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_9263_inc_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit0 @ X ) )
      = ( bit1 @ X ) ) ).

% inc.simps(2)
thf(fact_9264_inc_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit1 @ X ) )
      = ( bit0 @ ( inc @ X ) ) ) ).

% inc.simps(3)
thf(fact_9265_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_9266_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(5)
thf(fact_9267_inc__BitM__eq,axiom,
    ! [N: num] :
      ( ( inc @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% inc_BitM_eq
thf(fact_9268_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(9)
thf(fact_9269_BitM__inc__eq,axiom,
    ! [N: num] :
      ( ( bitM @ ( inc @ N ) )
      = ( bit1 @ N ) ) ).

% BitM_inc_eq
thf(fact_9270_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_9271_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_9272_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
      = ( bit1 @ M ) ) ).

% or_not_num_neg.simps(2)
thf(fact_9273_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(8)
thf(fact_9274_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_9275_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_9276_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_nat_less_exp
thf(fact_9277_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_9278_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N4: nat,M5: nat] : ( modulo_modulo_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% take_bit_nat_def
thf(fact_9279_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_9280_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa3: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X @ Xa3 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa3 = one )
           => ( Y != one ) ) )
       => ( ( ( X = one )
           => ! [M3: num] :
                ( ( Xa3
                  = ( bit0 @ M3 ) )
               => ( Y
                 != ( bit1 @ M3 ) ) ) )
         => ( ( ( X = one )
             => ! [M3: num] :
                  ( ( Xa3
                    = ( bit1 @ M3 ) )
                 => ( Y
                   != ( bit1 @ M3 ) ) ) )
           => ( ( ? [N2: num] :
                    ( X
                    = ( bit0 @ N2 ) )
               => ( ( Xa3 = one )
                 => ( Y
                   != ( bit0 @ one ) ) ) )
             => ( ! [N2: num] :
                    ( ( X
                      = ( bit0 @ N2 ) )
                   => ! [M3: num] :
                        ( ( Xa3
                          = ( bit0 @ M3 ) )
                       => ( Y
                         != ( bitM @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) )
               => ( ! [N2: num] :
                      ( ( X
                        = ( bit0 @ N2 ) )
                     => ! [M3: num] :
                          ( ( Xa3
                            = ( bit1 @ M3 ) )
                         => ( Y
                           != ( bit0 @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) )
                 => ( ( ? [N2: num] :
                          ( X
                          = ( bit1 @ N2 ) )
                     => ( ( Xa3 = one )
                       => ( Y != one ) ) )
                   => ( ! [N2: num] :
                          ( ( X
                            = ( bit1 @ N2 ) )
                         => ! [M3: num] :
                              ( ( Xa3
                                = ( bit0 @ M3 ) )
                             => ( Y
                               != ( bitM @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) )
                     => ~ ! [N2: num] :
                            ( ( X
                              = ( bit1 @ N2 ) )
                           => ! [M3: num] :
                                ( ( Xa3
                                  = ( bit1 @ M3 ) )
                               => ( Y
                                 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_9281_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_int_less_exp
thf(fact_9282_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N4: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% take_bit_int_def
thf(fact_9283_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_9284_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_9285_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_9286_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_9287_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_9288_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_9289_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_9290_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).

% take_bit_incr_eq
thf(fact_9291_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_minus_bit0
thf(fact_9292_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_9293_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_9294_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_9295_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_9296_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_9297_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
            @ ^ [Z7: complex] :
                ( ( power_power_complex @ Z7 @ N )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z7: complex] :
                ( ( power_power_complex @ Z7 @ N )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9298_real__root__zero,axiom,
    ! [N: nat] :
      ( ( root @ N @ zero_zero_real )
      = zero_zero_real ) ).

% real_root_zero
thf(fact_9299_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_9300_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_9301_root__0,axiom,
    ! [X: real] :
      ( ( root @ zero_zero_nat @ X )
      = zero_zero_real ) ).

% root_0
thf(fact_9302_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_9303_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_9304_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_9305_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_9306_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_9307_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_9308_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_9309_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_9310_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_9311_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_9312_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_9313_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_9314_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_9315_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_9316_real__root__minus,axiom,
    ! [N: nat,X: real] :
      ( ( root @ N @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_real @ ( root @ N @ X ) ) ) ).

% real_root_minus
thf(fact_9317_real__root__commute,axiom,
    ! [M: nat,N: nat,X: real] :
      ( ( root @ M @ ( root @ N @ X ) )
      = ( root @ N @ ( root @ M @ X ) ) ) ).

% real_root_commute
thf(fact_9318_real__root__mult__exp,axiom,
    ! [M: nat,N: nat,X: real] :
      ( ( root @ ( times_times_nat @ M @ N ) @ X )
      = ( root @ M @ ( root @ N @ X ) ) ) ).

% real_root_mult_exp
thf(fact_9319_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_9320_real__root__divide,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( root @ N @ ( divide_divide_real @ X @ Y ) )
      = ( divide_divide_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).

% real_root_divide
thf(fact_9321_real__root__inverse,axiom,
    ! [N: nat,X: real] :
      ( ( root @ N @ ( inverse_inverse_real @ X ) )
      = ( inverse_inverse_real @ ( root @ N @ X ) ) ) ).

% real_root_inverse
thf(fact_9322_real__root__pos__pos__le,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).

% real_root_pos_pos_le
thf(fact_9323_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_9324_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_9325_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( power_power_real @ X @ K ) )
        = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).

% real_root_power
thf(fact_9326_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_9327_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_9328_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_9329_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9330_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_9331_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_9332_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_9333_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_9334_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_9335_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_9336_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_9337_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_9338_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_9339_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_9340_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_9341_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_9342_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_9343_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_9344_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_9345_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L3: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ 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 @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_9346_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L3: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ 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 @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_9347_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L3: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9348_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N4: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N4 = zero_zero_nat )
            | ( ord_less_nat @ M5 @ N4 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M5 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M5 @ N4 ) @ N4 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_9349_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N4: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N4 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N4 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_9350_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_9351_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
thf(fact_9352_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q4: int,R5: int] : ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_9353_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N4: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M5 @ N4 ) @ ( modulo_modulo_nat @ M5 @ N4 ) ) ) ) ).

% divmod_nat_def
thf(fact_9354_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
      = L ) ).

% plus_integer_code(2)
thf(fact_9355_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% plus_integer_code(1)
thf(fact_9356_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_9357_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z7: int] :
                ( ( ord_less_eq_int @ D @ Z8 )
                & ( ord_less_int @ Z8 @ Z7 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_9358_divide__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( divide_divide_int @ Xa3 @ X ) ) ) ).

% divide_integer.abs_eq
thf(fact_9359_unset__bit__integer_Oabs__eq,axiom,
    ! [Xa3: nat,X: int] :
      ( ( bit_se8260200283734997820nteger @ Xa3 @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se4203085406695923979it_int @ Xa3 @ X ) ) ) ).

% unset_bit_integer.abs_eq
thf(fact_9360_zero__integer__def,axiom,
    ( zero_z3403309356797280102nteger
    = ( code_integer_of_int @ zero_zero_int ) ) ).

% zero_integer_def
thf(fact_9361_less__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
      = ( ord_less_int @ Xa3 @ X ) ) ).

% less_integer.abs_eq
thf(fact_9362_plus__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: int] :
      ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( plus_plus_int @ Xa3 @ X ) ) ) ).

% plus_integer.abs_eq
thf(fact_9363_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z7: int] :
                ( ( ord_less_eq_int @ D @ Z7 )
                & ( ord_less_int @ Z8 @ Z7 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_9364_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_9365_arctan__def,axiom,
    ( arctan
    = ( ^ [Y5: real] :
          ( the_real
          @ ^ [X2: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
              & ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X2 )
                = Y5 ) ) ) ) ) ).

% arctan_def
thf(fact_9366_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y5: real] :
          ( the_real
          @ ^ [X2: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
              & ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X2 )
                = Y5 ) ) ) ) ) ).

% arcsin_def
thf(fact_9367_nat__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
      = N ) ).

% nat_int
thf(fact_9368_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_9369_nat__of__bool,axiom,
    ! [P: $o] :
      ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% nat_of_bool
thf(fact_9370_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_9371_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_9372_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_9373_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_9374_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_9375_nat__zminus__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_9376_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = Z ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_9377_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_9378_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_9379_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_9380_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_9381_nat__abs__dvd__iff,axiom,
    ! [K: int,N: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
      = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_abs_dvd_iff
thf(fact_9382_dvd__nat__abs__iff,axiom,
    ! [N: nat,K: int] :
      ( ( dvd_dvd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
      = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).

% dvd_nat_abs_iff
thf(fact_9383_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_9384_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_9385_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_9386_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_9387_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_9388_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_9389_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_9390_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I2: num] : ( nat2 @ ( numeral_numeral_int @ I2 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_9391_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_9392_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_9393_ex__nat,axiom,
    ( ( ^ [P5: nat > $o] :
        ? [X7: nat] : ( P5 @ X7 ) )
    = ( ^ [P6: nat > $o] :
        ? [X2: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X2 )
          & ( P6 @ ( nat2 @ X2 ) ) ) ) ) ).

% ex_nat
thf(fact_9394_all__nat,axiom,
    ( ( ^ [P5: nat > $o] :
        ! [X7: nat] : ( P5 @ X7 ) )
    = ( ^ [P6: nat > $o] :
        ! [X2: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X2 )
         => ( P6 @ ( nat2 @ X2 ) ) ) ) ) ).

% all_nat
thf(fact_9395_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
       => ( ( ( nat2 @ Z )
            = ( nat2 @ Z4 ) )
          = ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_9396_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M5: nat,N4: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M5 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_9397_nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( bit_se2002935070580805687sk_nat @ N ) ) ).

% nat_mask_eq
thf(fact_9398_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_9399_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).

% zless_nat_eq_int_zless
thf(fact_9400_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_9401_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_9402_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = Z )
      = ( ( M
          = ( nat2 @ Z ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).

% int_eq_iff
thf(fact_9403_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_9404_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_9405_real__nat__ceiling__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_9406_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_9407_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N4: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% and_nat_def
thf(fact_9408_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N4: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% or_nat_def
thf(fact_9409_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A4: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_9410_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A4: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_9411_ln__neg__is__const,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ln_ln_real @ X )
        = ( the_real
          @ ^ [X2: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9412_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_9413_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M
        = ( nat2 @ W ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
         => ( W
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
thf(fact_9414_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( ( nat2 @ W )
        = M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
         => ( W
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
thf(fact_9415_split__nat,axiom,
    ! [P: nat > $o,I: int] :
      ( ( P @ ( nat2 @ I ) )
      = ( ! [N4: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N4 ) )
           => ( P @ N4 ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_9416_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_9417_nat__add__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z4 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_9418_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).

% le_nat_iff
thf(fact_9419_Suc__as__int,axiom,
    ( suc
    = ( ^ [A4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_9420_nat__mult__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z4 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) ) ) ) ).

% nat_mult_distrib
thf(fact_9421_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_9422_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
     => ( ( ord_less_eq_int @ Z4 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z4 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_9423_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_9424_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_9425_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_9426_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_9427_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_9428_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_9429_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_9430_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% mod_abs_eq_div_nat
thf(fact_9431_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
        = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_9432_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_9433_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_9434_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_9435_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_9436_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_9437_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
        = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% nat_less_iff
thf(fact_9438_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z4 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z4 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_9439_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_9440_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_9441_diff__nat__eq__if,axiom,
    ! [Z4: int,Z: int] :
      ( ( ( ord_less_int @ Z4 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z4 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z4 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z4 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_9442_arccos__def,axiom,
    ( arccos
    = ( ^ [Y5: real] :
          ( the_real
          @ ^ [X2: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X2 )
              & ( ord_less_eq_real @ X2 @ pi )
              & ( ( cos_real @ X2 )
                = Y5 ) ) ) ) ) ).

% arccos_def
thf(fact_9443_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_9444_even__nat__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).

% even_nat_iff
thf(fact_9445_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X2: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X2 )
          & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X2 )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9446_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X2: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X2 )
            & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X2 )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9447_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
            = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
            = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_9448_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K3: int,L3: int] :
          ( if_int @ ( L3 = zero_zero_int ) @ K3
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L3 ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L3 )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L3 )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L3 @ K3 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_9449_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K3: int,L3: int] :
          ( if_int @ ( L3 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L3 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L3 @ K3 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9450_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_9451_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_9452_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_9453_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_9454_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L4: int] :
          ( K
         != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% int_sgnE
thf(fact_9455_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( ( sgn_sgn_int @ K )
        = ( sgn_sgn_int @ L ) )
     => ( ( divide_divide_int @ K @ L )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_eq_sgn_abs
thf(fact_9456_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ L @ K )
       => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
          = ( sgn_sgn_int @ L ) ) ) ) ).

% sgn_mod
thf(fact_9457_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I2: int] : ( if_int @ ( I2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_9458_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_9459_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( dvd_dvd_int @ L @ K )
     => ( ( divide_divide_int @ K @ L )
        = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_dvd_sgn_abs
thf(fact_9460_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q2: int] :
      ( ( ( sgn_sgn_int @ R2 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q2 @ L ) @ R2 ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_9461_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( ( sgn_sgn_int @ K )
         != ( sgn_sgn_int @ L ) )
       => ( ( divide_divide_int @ K @ L )
          = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_9462_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 ) ) )
          | ? [L3: int,K3: int,Q4: int] :
              ( ( A12 = K3 )
              & ( A23 = L3 )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
              & ( L3 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q4 @ L3 ) ) )
          | ? [R5: int,L3: int,K3: int,Q4: int] :
              ( ( A12 = K3 )
              & ( A23 = L3 )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L3 ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L3 ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q4 @ L3 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_9463_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 ) ) )
       => ( ! [Q3: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
             => ( ( A22 != zero_zero_int )
               => ( A1
                 != ( times_times_int @ Q3 @ A22 ) ) ) )
         => ~ ! [R: int,Q3: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q3 @ R ) )
               => ( ( ( sgn_sgn_int @ R )
                    = ( sgn_sgn_int @ A22 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ A22 ) )
                   => ( A1
                     != ( plus_plus_int @ ( times_times_int @ Q3 @ A22 ) @ R ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_9464_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_9465_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_9466_sgn__div__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( divide_divide_int @ A @ B )
       != zero_zero_int )
     => ( ( sgn_sgn_int @ ( divide_divide_int @ A @ B ) )
        = ( sgn_sgn_int @ ( times_times_int @ A @ B ) ) ) ) ).

% sgn_div_eq_sgn_mult
thf(fact_9467_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_9468_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_sgn_iff
thf(fact_9469_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_9470_not__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% not_negative_int_iff
thf(fact_9471_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% not_nonnegative_int_iff
thf(fact_9472_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_9473_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N ) ) ).

% signed_take_bit_negative_iff
thf(fact_9474_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_9475_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_9476_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_9477_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_9478_bin__nth__minus__Bit0,axiom,
    ! [N: nat,W: num] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ N )
        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% bin_nth_minus_Bit0
thf(fact_9479_bin__nth__minus__Bit1,axiom,
    ! [N: nat,W: num] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ N )
        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% bin_nth_minus_Bit1
thf(fact_9480_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_9481_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_9482_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
      = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).

% bit_minus_int_iff
thf(fact_9483_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_not_int_iff
thf(fact_9484_bit__and__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
        & ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).

% bit_and_int_iff
thf(fact_9485_bit__or__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
        | ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).

% bit_or_int_iff
thf(fact_9486_real__sgn__eq,axiom,
    ( sgn_sgn_real
    = ( ^ [X2: real] : ( divide_divide_real @ X2 @ ( abs_abs_real @ X2 ) ) ) ) ).

% real_sgn_eq
thf(fact_9487_or__int__def,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L3: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L3 ) ) ) ) ) ).

% or_int_def
thf(fact_9488_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_9489_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_9490_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_9491_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_9492_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_not_int_iff'
thf(fact_9493_sgn__eq,axiom,
    ( sgn_sgn_complex
    = ( ^ [Z7: complex] : ( divide1717551699836669952omplex @ Z7 @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ Z7 ) ) ) ) ) ).

% sgn_eq
thf(fact_9494_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_9495_not__int__div__2,axiom,
    ! [K: int] :
      ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% not_int_div_2
thf(fact_9496_even__not__iff__int,axiom,
    ! [K: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).

% even_not_iff_int
thf(fact_9497_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_9498_and__not__numerals_I4_J,axiom,
    ! [M: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).

% and_not_numerals(4)
thf(fact_9499_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_9500_or__not__numerals_I4_J,axiom,
    ! [M: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).

% or_not_numerals(4)
thf(fact_9501_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_9502_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).

% int_numeral_or_not_num_neg
thf(fact_9503_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).

% int_numeral_not_or_num_neg
thf(fact_9504_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
      = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% numeral_or_not_num_eq
thf(fact_9505_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_9506_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ N2 @ M2 )
             => ( ( bit_se1146084159140164899it_int @ K @ M2 )
                = ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).

% int_bit_bound
thf(fact_9507_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(5)
thf(fact_9508_and__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).

% and_not_numerals(7)
thf(fact_9509_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_9510_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_9511_or__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(7)
thf(fact_9512_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(6)
thf(fact_9513_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(9)
thf(fact_9514_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% or_not_numerals(6)
thf(fact_9515_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_9516_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_9517_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K3: int,N4: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% bit_int_def
thf(fact_9518_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( ( sgn_sgn_complex @ Z )
        = ( cis @ X ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ( arg @ Z )
            = X ) ) ) ) ).

% cis_Arg_unique
thf(fact_9519_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 )
         => ! [Y5: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N ) )
                = X )
             => ( P @ Y5 ) ) ) ) ) ).

% split_root
thf(fact_9520_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( 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 @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_9521_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( ( sgn_sgn_complex @ Z )
          = ( cis @ ( arg @ Z ) ) )
        & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
        & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).

% Arg_correct
thf(fact_9522_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( 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 @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_9523_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( 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 @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_9524_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( 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 @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_9525_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_9526_Bit__Operations_Oset__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N4: nat,K3: int] :
          ( plus_plus_int @ K3
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K3 @ N4 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% Bit_Operations.set_bit_eq
thf(fact_9527_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% unset_bit_eq
thf(fact_9528_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_9529_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_9530_int__not__code_I1_J,axiom,
    ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% int_not_code(1)
thf(fact_9531_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L3: int] :
          ( if_int
          @ ( K3
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L3 )
          @ ( if_int
            @ ( L3
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K3 )
            @ ( if_int @ ( K3 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = 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 @ L3 @ ( 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 @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9532_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_9533_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
       != ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_9534_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_9535_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_9536_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
       != ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).

% bit_xor_int_iff
thf(fact_9537_int__xor__code_I1_J,axiom,
    ! [J2: int] :
      ( ( bit_se6526347334894502574or_int @ zero_zero_int @ J2 )
      = J2 ) ).

% int_xor_code(1)
thf(fact_9538_int__xor__code_I2_J,axiom,
    ! [I: int] :
      ( ( bit_se6526347334894502574or_int @ I @ zero_zero_int )
      = I ) ).

% int_xor_code(2)
thf(fact_9539_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_9540_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_9541_xor__int__def,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L3: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L3 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L3 ) ) ) ) ).

% xor_int_def
thf(fact_9542_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_nat_iff
thf(fact_9543_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M5: nat,N4: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% bit_nat_def
thf(fact_9544_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_9545_test__bit__int__code_I1_J,axiom,
    ! [N: nat] :
      ~ ( bit_se1146084159140164899it_int @ zero_zero_int @ N ) ).

% test_bit_int_code(1)
thf(fact_9546_int__and__code_I2_J,axiom,
    ! [I: int] :
      ( ( bit_se725231765392027082nd_int @ I @ zero_zero_int )
      = zero_zero_int ) ).

% int_and_code(2)
thf(fact_9547_int__and__code_I1_J,axiom,
    ! [J2: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ J2 )
      = zero_zero_int ) ).

% int_and_code(1)
thf(fact_9548_int__or__code_I1_J,axiom,
    ! [J2: int] :
      ( ( bit_se1409905431419307370or_int @ zero_zero_int @ J2 )
      = J2 ) ).

% int_or_code(1)
thf(fact_9549_int__or__code_I2_J,axiom,
    ! [I: int] :
      ( ( bit_se1409905431419307370or_int @ I @ zero_zero_int )
      = I ) ).

% int_or_code(2)
thf(fact_9550_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L3: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_9551_Arg__def,axiom,
    ( arg
    = ( ^ [Z7: complex] :
          ( if_real @ ( Z7 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A4: real] :
                ( ( ( sgn_sgn_complex @ Z7 )
                  = ( cis @ A4 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
                & ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9552_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_9553_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_9554_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_9555_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_9556_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N4: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% xor_nat_def
thf(fact_9557_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9558_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9559_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_9560_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_9561_int__lsb__numeral_I1_J,axiom,
    ~ ( least_4859182151741483524sb_int @ zero_zero_int ) ).

% int_lsb_numeral(1)
thf(fact_9562_int__lsb__numeral_I6_J,axiom,
    ! [W: num] :
      ~ ( least_4859182151741483524sb_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) ).

% int_lsb_numeral(6)
thf(fact_9563_int__lsb__numeral_I3_J,axiom,
    least_4859182151741483524sb_int @ ( numeral_numeral_int @ one ) ).

% int_lsb_numeral(3)
thf(fact_9564_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : ( least_4859182151741483524sb_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) ).

% int_lsb_numeral(7)
thf(fact_9565_int__lsb__numeral_I8_J,axiom,
    ! [W: num] :
      ~ ( least_4859182151741483524sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) ) ).

% int_lsb_numeral(8)
thf(fact_9566_int__lsb__numeral_I5_J,axiom,
    least_4859182151741483524sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ).

% int_lsb_numeral(5)
thf(fact_9567_int__lsb__numeral_I9_J,axiom,
    ! [W: num] : ( least_4859182151741483524sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) ) ).

% int_lsb_numeral(9)
thf(fact_9568_lsb__int__def,axiom,
    ( least_4859182151741483524sb_int
    = ( ^ [I2: int] : ( bit_se1146084159140164899it_int @ I2 @ zero_zero_nat ) ) ) ).

% lsb_int_def
thf(fact_9569_bin__last__conv__lsb,axiom,
    ( ( ^ [A4: int] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) )
    = least_4859182151741483524sb_int ) ).

% bin_last_conv_lsb
thf(fact_9570_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V3352910403632780892pi_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = one_one_int )
             => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y = one_one_int )
               => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
thf(fact_9571_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_9572_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V5144397997797733112_d_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
             => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
               => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
thf(fact_9573_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V1247956027447740395_p_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
thf(fact_9574_VEBT__internal_OTb_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_VEBT_Tb @ X )
        = Y )
     => ( ( accp_nat @ vEBT_VEBT_Tb_rel2 @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_int @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_int @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
thf(fact_9575_VEBT__internal_OTb_H_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X )
        = Y )
     => ( ( accp_nat @ vEBT_VEBT_Tb_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
thf(fact_9576_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V2957053500504383685pi_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( suc @ zero_zero_nat ) )
             => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( suc @ zero_zero_nat ) )
               => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
thf(fact_9577_vebt__minti_Opelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_minti @ X )
        = Y )
     => ( ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ X )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leafi @ A6 @ B5 ) )
             => ( ( ( A6
                   => ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                  & ( ~ A6
                   => ( ( B5
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                      & ( ~ B5
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) )
               => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Leafi @ A6 @ B5 ) ) ) )
         => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( ( X
                  = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y
                    = ( heap_T3487192422709364219on_nat @ none_nat ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
                  ( ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
                 => ( ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi2 ) ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ) ) ) ) ).

% vebt_minti.pelims
thf(fact_9578_vebt__maxti_Opelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_maxti @ X )
        = Y )
     => ( ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ X )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leafi @ A6 @ B5 ) )
             => ( ( ( B5
                   => ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                  & ( ~ B5
                   => ( ( A6
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                      & ( ~ A6
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) )
               => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Leafi @ A6 @ B5 ) ) ) )
         => ( ! [Uu: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( ( X
                  = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y
                    = ( heap_T3487192422709364219on_nat @ none_nat ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
                  ( ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
                 => ( ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma2 ) ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ) ) ) ) ).

% vebt_maxti.pelims
thf(fact_9579_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_9580_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_9581_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_9582_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_9583_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo2: option4927543243414619207at_nat,Deg: nat,Tree_list2: list_VEBT_VEBT,Summary: vEBT_VEBT,Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ Mmo2 @ Deg @ Tree_list2 @ Summary ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
      = ( times_times_assn
        @ ( times_times_assn
          @ ( pure_assn
            @ ( ( Mmoi2 = Mmo2 )
              & ( Degi2 = Deg ) ) )
          @ ( vEBT_vebt_assn_raw @ Summary @ Summaryi2 ) )
        @ ( ex_ass463751140784270563_VEBTi
          @ ^ [Tree_is2: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array2 @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list2 @ Tree_is2 ) ) ) ) ) ).

% vebt_assn_raw.simps(2)
thf(fact_9584_VEBT__internal_Ospace_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ X )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A6 @ B5 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList2 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.space.pelims
thf(fact_9585_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space2 @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ X )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A6 @ B5 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList2 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
thf(fact_9586_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( ( vEBT_VEBT_cnt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Y = one_one_real )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Leaf @ A6 @ B5 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList2 ) @ zero_zero_real ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
thf(fact_9587_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ X )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A6 @ B5 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList2 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
thf(fact_9588_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9589_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9590_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_9591_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_9592_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_9593_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_numeral_minus_bit0
thf(fact_9594_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_9595_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_numeral_minus_bit1
thf(fact_9596_drop__bit__int__code_I2_J,axiom,
    ! [N: nat] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ zero_zero_int )
      = zero_zero_int ) ).

% drop_bit_int_code(2)
thf(fact_9597_drop__bit__int__code_I1_J,axiom,
    ! [I: int] :
      ( ( bit_se8568078237143864401it_int @ zero_zero_nat @ I )
      = I ) ).

% drop_bit_int_code(1)
thf(fact_9598_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).

% drop_bit_nat_eq
thf(fact_9599_bin__rest__code,axiom,
    ! [I: int] :
      ( ( divide_divide_int @ I @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se8568078237143864401it_int @ one_one_nat @ I ) ) ).

% bin_rest_code
thf(fact_9600_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N4: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% drop_bit_int_def
thf(fact_9601_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N4: nat,M5: nat] : ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9602_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( ( B5
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B5
                   => ( ( A6
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A6
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A6 @ B5 ) ) ) )
         => ( ! [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,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
                 => ( ( Y
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_9603_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( ( A6
                   => ( Y
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A6
                   => ( ( B5
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B5
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A6 @ B5 ) ) ) )
         => ( ! [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,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
                 => ( ( Y
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_9604_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A6 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A6 @ B5 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_9605_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B5 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Leaf @ A6 @ B5 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_9606_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( ( Y = one_one_nat )
             => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
           => ( ! [Uu: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu @ $true ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu @ $true ) ) ) )
             => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ) )
               => ~ ! [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va3 @ Vb @ Vc2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va3 @ Vb @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
thf(fact_9607_merge__true__star,axiom,
    ( ( times_times_assn @ top_top_assn @ top_top_assn )
    = top_top_assn ) ).

% merge_true_star
thf(fact_9608_ent__true,axiom,
    ! [P: assn] : ( entails @ P @ top_top_assn ) ).

% ent_true
thf(fact_9609_merge__true__star__ctx,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ top_top_assn @ ( times_times_assn @ top_top_assn @ P ) )
      = ( times_times_assn @ top_top_assn @ P ) ) ).

% merge_true_star_ctx
thf(fact_9610_ent__star__mono__true,axiom,
    ! [A2: assn,A7: assn,B3: assn,B8: assn] :
      ( ( entails @ A2 @ ( times_times_assn @ A7 @ top_top_assn ) )
     => ( ( entails @ B3 @ ( times_times_assn @ B8 @ top_top_assn ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ A2 @ B3 ) @ top_top_assn ) @ ( times_times_assn @ ( times_times_assn @ A7 @ B8 ) @ top_top_assn ) ) ) ) ).

% ent_star_mono_true
thf(fact_9611_ent__refl__true,axiom,
    ! [A2: assn] : ( entails @ A2 @ ( times_times_assn @ A2 @ top_top_assn ) ) ).

% ent_refl_true
thf(fact_9612_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R3: assn] :
      ( ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) )
     => ( entails @ ( times_times_assn @ P @ R3 ) @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% ent_true_drop(1)
thf(fact_9613_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% ent_true_drop(2)
thf(fact_9614_mod__star__trueI,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P @ H2 )
     => ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H2 ) ) ).

% mod_star_trueI
thf(fact_9615_mod__star__trueE,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H2 )
     => ~ ! [H4: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P @ H4 ) ) ).

% mod_star_trueE
thf(fact_9616_shiftr__integer__conv__div__pow2,axiom,
    ( bit_se3928097537394005634nteger
    = ( ^ [N4: nat,X2: code_integer] : ( divide6298287555418463151nteger @ X2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% shiftr_integer_conv_div_pow2
thf(fact_9617_vebt__assn__raw_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa3: vEBT_VEBTi,Y: assn] :
      ( ( ( vEBT_vebt_assn_raw @ X @ Xa3 )
        = Y )
     => ( ! [A6: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xa3
                  = ( vEBT_Leafi @ Ai @ Bi ) )
               => ( Y
                 != ( pure_assn
                    @ ( ( Ai = A6 )
                      & ( Bi = B5 ) ) ) ) ) )
       => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) )
             => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                  ( ( Xa3
                    = ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
                 => ( Y
                   != ( times_times_assn
                      @ ( times_times_assn
                        @ ( pure_assn
                          @ ( ( Mmoi = Mmo )
                            & ( Degi = Deg2 ) ) )
                        @ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi ) )
                      @ ( ex_ass463751140784270563_VEBTi
                        @ ^ [Tree_is2: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list @ Tree_is2 ) ) ) ) ) ) )
         => ( ( ? [V2: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ V2 @ Va @ Vb3 @ Vc3 ) )
             => ( ? [Vd3: $o,Ve3: $o] :
                    ( Xa3
                    = ( vEBT_Leafi @ Vd3 @ Ve3 ) )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] :
                    ( X
                    = ( vEBT_Leaf @ Vd3 @ Ve3 ) )
               => ( ? [V2: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                      ( Xa3
                      = ( vEBT_Nodei @ V2 @ Va @ Vb3 @ Vc3 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
thf(fact_9618_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,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
                   => ( Y
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ) )
               => ~ ! [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va3 @ Vb @ Vc2 ) )
                     => ( ~ Y
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va3 @ Vb @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_9619_star__false__right,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ P @ bot_bot_assn )
      = bot_bot_assn ) ).

% star_false_right
thf(fact_9620_star__false__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ bot_bot_assn @ P )
      = bot_bot_assn ) ).

% star_false_left
thf(fact_9621_ent__false__iff,axiom,
    ! [P: assn] :
      ( ( entails @ P @ bot_bot_assn )
      = ( ! [H: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P @ H ) ) ) ).

% ent_false_iff
thf(fact_9622_infinite__UNIV__int,axiom,
    ~ ( finite_finite_int @ top_top_set_int ) ).

% infinite_UNIV_int
thf(fact_9623_ent__false,axiom,
    ! [P: assn] : ( entails @ bot_bot_assn @ P ) ).

% ent_false
thf(fact_9624_lsb__integer__code,axiom,
    ( least_7544222001954398261nteger
    = ( ^ [X2: code_integer] : ( bit_se9216721137139052372nteger @ X2 @ zero_zero_nat ) ) ) ).

% lsb_integer_code
thf(fact_9625_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option4927543243414619207at_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc: vEBT_VEBT,Vd: $o,Ve: $o] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ V @ Va2 @ Vb2 @ Vc ) @ ( vEBT_Leafi @ Vd @ Ve ) )
      = bot_bot_assn ) ).

% vebt_assn_raw.simps(3)
thf(fact_9626_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd: $o,Ve: $o,V: option4927543243414619207at_nat,Va2: nat,Vb2: array_VEBT_VEBTi,Vc: vEBT_VEBTi] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ Vd @ Ve ) @ ( vEBT_Nodei @ V @ Va2 @ Vb2 @ Vc ) )
      = bot_bot_assn ) ).

% vebt_assn_raw.simps(4)
thf(fact_9627_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 ) ) )
           => ~ ! [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va3 @ Vb @ Vc2 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va3 @ Vb @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_9628_Bit__integer_Oabs__eq,axiom,
    ! [Xa3: int,X: $o] :
      ( ( bits_Bit_integer @ ( code_integer_of_int @ Xa3 ) @ X )
      = ( code_integer_of_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ X ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa3 ) ) ) ) ).

% Bit_integer.abs_eq
thf(fact_9629_vebt__assn__raw_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa3: vEBT_VEBTi,Y: assn] :
      ( ( ( vEBT_vebt_assn_raw @ X @ Xa3 )
        = Y )
     => ( ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ X @ Xa3 ) )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xa3
                    = ( vEBT_Leafi @ Ai @ Bi ) )
                 => ( ( Y
                      = ( pure_assn
                        @ ( ( Ai = A6 )
                          & ( Bi = B5 ) ) ) )
                   => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ A6 @ B5 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) ) ) ) )
         => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) )
               => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                    ( ( Xa3
                      = ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
                   => ( ( Y
                        = ( times_times_assn
                          @ ( times_times_assn
                            @ ( pure_assn
                              @ ( ( Mmoi = Mmo )
                                & ( Degi = Deg2 ) ) )
                            @ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi ) )
                          @ ( ex_ass463751140784270563_VEBTi
                            @ ^ [Tree_is2: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list @ Tree_is2 ) ) ) ) )
                     => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) ) ) ) ) )
           => ( ! [V2: option4927543243414619207at_nat,Va: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ V2 @ Va @ Vb3 @ Vc3 ) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xa3
                        = ( vEBT_Leafi @ Vd3 @ Ve3 ) )
                     => ( ( Y = bot_bot_assn )
                       => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ V2 @ Va @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) ) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( X
                      = ( vEBT_Leaf @ Vd3 @ Ve3 ) )
                   => ! [V2: option4927543243414619207at_nat,Va: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                        ( ( Xa3
                          = ( vEBT_Nodei @ V2 @ Va @ Vb3 @ Vc3 ) )
                       => ( ( Y = bot_bot_assn )
                         => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V2 @ Va @ Vb3 @ Vc3 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
thf(fact_9630_root__def,axiom,
    ( root
    = ( ^ [N4: nat,X2: real] :
          ( if_real @ ( N4 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y5: real] : ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N4 ) )
            @ X2 ) ) ) ) ).

% root_def
thf(fact_9631_dup__1,axiom,
    ( ( code_dup @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% dup_1
thf(fact_9632_bin__last__integer__nbe,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [I2: code_integer] :
          ( ( modulo364778990260209775nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) ) ) ).

% bin_last_integer_nbe
thf(fact_9633_dup_Oabs__eq,axiom,
    ! [X: int] :
      ( ( code_dup @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( plus_plus_int @ X @ X ) ) ) ).

% dup.abs_eq
thf(fact_9634_bin__last__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( bits_b8758750999018896077nteger @ ( code_integer_of_int @ X ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) ) ) ).

% bin_last_integer.abs_eq
thf(fact_9635_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_9636_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_9637_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_9638_push__bit__int__code_I1_J,axiom,
    ! [I: int] :
      ( ( bit_se545348938243370406it_int @ zero_zero_nat @ I )
      = I ) ).

% push_bit_int_code(1)
thf(fact_9639_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).

% push_bit_nat_eq
thf(fact_9640_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
      = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).

% drop_bit_push_bit_int
thf(fact_9641_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M5: nat,N4: nat] : ( bit_se1412395901928357646or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M5 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9642_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M5: nat,N4: nat] : ( bit_se6528837805403552850or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M5 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_9643_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9644_Bit__Operations_Oset__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N4: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).

% Bit_Operations.set_bit_int_def
thf(fact_9645_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9646_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N4: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_9647_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N4: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_9648_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N4: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% push_bit_int_def
thf(fact_9649_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] :
      ( ( bits_Bit_integer @ I @ $true )
      = ( plus_p5714425477246183910nteger @ ( bit_se7788150548672797655nteger @ one_one_nat @ I ) @ one_one_Code_integer ) ) ).

% Bit_integer_code(2)
thf(fact_9650_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N4: nat,M5: nat] : ( times_times_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% push_bit_nat_def
thf(fact_9651_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_9652_shiftl__integer__conv__mult__pow2,axiom,
    ( bit_se7788150548672797655nteger
    = ( ^ [N4: nat,X2: code_integer] : ( times_3573771949741848930nteger @ X2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% shiftl_integer_conv_mult_pow2
thf(fact_9653_bin__rest__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( bits_b2549910563261871055nteger @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% bin_rest_integer.abs_eq
thf(fact_9654_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I2: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9655_bin__rest__integer__code,axiom,
    ( bits_b2549910563261871055nteger
    = ( ^ [I2: code_integer] : ( divide6298287555418463151nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% bin_rest_integer_code
thf(fact_9656_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
      = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_9657_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_9658_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] :
      ( ( bit_concat_bit @ N @ K @ zero_zero_int )
      = ( bit_se2923211474154528505it_int @ N @ K ) ) ).

% concat_bit_of_zero_2
thf(fact_9659_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
      = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_9660_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
      = ( ord_less_int @ L @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_9661_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_9662_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L: int,R2: int] :
      ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R2 ) ) ).

% concat_bit_assoc
thf(fact_9663_concat__bit__eq__iff,axiom,
    ! [N: nat,K: int,L: int,R2: int,S: int] :
      ( ( ( bit_concat_bit @ N @ K @ L )
        = ( bit_concat_bit @ N @ R2 @ S ) )
      = ( ( ( bit_se2923211474154528505it_int @ N @ K )
          = ( bit_se2923211474154528505it_int @ N @ R2 ) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
thf(fact_9664_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_9665_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N4: nat,K3: int,L3: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N4 @ K3 ) @ ( bit_se545348938243370406it_int @ N4 @ L3 ) ) ) ) ).

% concat_bit_eq
thf(fact_9666_concat__bit__def,axiom,
    ( bit_concat_bit
    = ( ^ [N4: nat,K3: int,L3: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N4 @ K3 ) @ ( bit_se545348938243370406it_int @ N4 @ L3 ) ) ) ) ).

% concat_bit_def
thf(fact_9667_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
      = ( ( ( ord_less_nat @ N @ M )
          & ( bit_se1146084159140164899it_int @ K @ N ) )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9668_signed__take__bit__eq__concat__bit,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,K3: int] : ( bit_concat_bit @ N4 @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) ) ) ) ) ).

% signed_take_bit_eq_concat_bit
thf(fact_9669_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_9670_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X8: nat > real] :
        ! [J3: nat] :
        ? [M9: nat] :
        ! [M5: nat] :
          ( ( ord_less_eq_nat @ M9 @ M5 )
         => ! [N4: nat] :
              ( ( ord_less_eq_nat @ M9 @ N4 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N4 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9671_sdiv__int__div__0,axiom,
    ! [X: int] :
      ( ( signed6714573509424544716de_int @ X @ zero_zero_int )
      = zero_zero_int ) ).

% sdiv_int_div_0
thf(fact_9672_sdiv__int__0__div,axiom,
    ! [X: int] :
      ( ( signed6714573509424544716de_int @ zero_zero_int @ X )
      = zero_zero_int ) ).

% sdiv_int_0_div
thf(fact_9673_int__sdiv__simps_I2_J,axiom,
    ! [A: int] :
      ( ( signed6714573509424544716de_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% int_sdiv_simps(2)
thf(fact_9674_int__sdiv__same__is__1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( ( signed6714573509424544716de_int @ A @ B )
          = A )
        = ( B = one_one_int ) ) ) ).

% int_sdiv_same_is_1
thf(fact_9675_sdiv__int__numeral__numeral,axiom,
    ! [M: num,N: num] :
      ( ( signed6714573509424544716de_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ).

% sdiv_int_numeral_numeral
thf(fact_9676_int__sdiv__negated__is__minus1,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( ( signed6714573509424544716de_int @ A @ B )
          = ( uminus_uminus_int @ A ) )
        = ( B
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% int_sdiv_negated_is_minus1
thf(fact_9677_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( signed6714573509424544716de_int @ A @ B )
       != zero_zero_int )
     => ( ( sgn_sgn_int @ ( signed6714573509424544716de_int @ A @ B ) )
        = ( sgn_sgn_int @ ( times_times_int @ A @ B ) ) ) ) ).

% sgn_sdiv_eq_sgn_mult
thf(fact_9678_signed__divide__int__def,axiom,
    ( signed6714573509424544716de_int
    = ( ^ [K3: int,L3: int] : ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( sgn_sgn_int @ L3 ) ) @ ( divide_divide_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L3 ) ) ) ) ) ).

% signed_divide_int_def
thf(fact_9679_entails__solve__init_I1_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( fI_QUERY @ P @ Q @ top_top_assn )
     => ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% entails_solve_init(1)
thf(fact_9680_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_9681_FI__QUERY__def,axiom,
    ( fI_QUERY
    = ( ^ [P6: assn,Q7: assn,F7: assn] : ( entails @ P6 @ ( times_times_assn @ Q7 @ F7 ) ) ) ) ).

% FI_QUERY_def
thf(fact_9682_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F2: assn] :
      ( ( fI_QUERY @ P @ Q @ F2 )
     => ( entails @ P @ ( times_times_assn @ Q @ F2 ) ) ) ).

% frame_inference_init
thf(fact_9683_entails__solve__init_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( fI_QUERY @ P @ Q @ one_one_assn )
     => ( entails @ P @ Q ) ) ).

% entails_solve_init(2)
thf(fact_9684_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_9685_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_9686_smod__int__range,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( member_int @ ( signed6292675348222524329lo_int @ A @ B ) @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ ( uminus_uminus_int @ ( abs_abs_int @ B ) ) @ one_one_int ) @ ( minus_minus_int @ ( abs_abs_int @ B ) @ one_one_int ) ) ) ) ).

% smod_int_range
thf(fact_9687_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_9688_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D2: nat] :
      ( ( vEBT_VEBT_valid @ T @ D2 )
     => ( vEBT_invar_vebt @ T @ D2 ) ) ).

% valid_eq2
thf(fact_9689_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D2: nat] :
      ( ( vEBT_invar_vebt @ T @ D2 )
     => ( vEBT_VEBT_valid @ T @ D2 ) ) ).

% valid_eq1
thf(fact_9690_smod__int__0__mod,axiom,
    ! [X: int] :
      ( ( signed6292675348222524329lo_int @ zero_zero_int @ X )
      = zero_zero_int ) ).

% smod_int_0_mod
thf(fact_9691_smod__int__mod__0,axiom,
    ! [X: int] :
      ( ( signed6292675348222524329lo_int @ X @ zero_zero_int )
      = X ) ).

% smod_int_mod_0
thf(fact_9692_smod__int__numeral__numeral,axiom,
    ! [M: num,N: num] :
      ( ( signed6292675348222524329lo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ).

% smod_int_numeral_numeral
thf(fact_9693_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,D2: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D2 )
      = ( D2 = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_9694_smod__int__compares_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ B @ ( signed6292675348222524329lo_int @ A @ B ) ) ) ) ).

% smod_int_compares(8)
thf(fact_9695_smod__int__compares_I7_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( signed6292675348222524329lo_int @ A @ B ) @ zero_zero_int ) ) ) ).

% smod_int_compares(7)
thf(fact_9696_smod__int__compares_I6_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( signed6292675348222524329lo_int @ A @ B ) ) ) ) ).

% smod_int_compares(6)
thf(fact_9697_smod__int__compares_I4_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( signed6292675348222524329lo_int @ A @ B ) @ zero_zero_int ) ) ) ).

% smod_int_compares(4)
thf(fact_9698_smod__int__compares_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 @ zero_zero_int @ ( signed6292675348222524329lo_int @ A @ B ) ) ) ) ).

% smod_int_compares(2)
thf(fact_9699_smod__int__compares_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( signed6292675348222524329lo_int @ A @ B ) @ B ) ) ) ).

% smod_int_compares(1)
thf(fact_9700_smod__mod__positive,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( signed6292675348222524329lo_int @ A @ B )
          = ( modulo_modulo_int @ A @ B ) ) ) ) ).

% smod_mod_positive
thf(fact_9701_smod__int__compares_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( signed6292675348222524329lo_int @ A @ B ) ) ) ) ).

% smod_int_compares(3)
thf(fact_9702_smod__int__compares_I5_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( signed6292675348222524329lo_int @ A @ B ) @ ( uminus_uminus_int @ B ) ) ) ) ).

% smod_int_compares(5)
thf(fact_9703_uint32_Osize__eq__length,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
    = ( type_l796852477590012082l_num1 @ type_N8448461349408098053l_num1 ) ) ).

% uint32.size_eq_length
thf(fact_9704_len__num0,axiom,
    ( type_l4264026598287037464l_num0
    = ( ^ [Uu4: itself_Numeral_num0] : zero_zero_nat ) ) ).

% len_num0
thf(fact_9705_len__of__finite__2__def,axiom,
    ( type_l31302759751748492nite_2
    = ( ^ [X2: itself_finite_2] : ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% len_of_finite_2_def
thf(fact_9706_len__of__finite__3__def,axiom,
    ( type_l31302759751748493nite_3
    = ( ^ [X2: itself_finite_3] : ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% len_of_finite_3_def
thf(fact_9707_bij__betw__Suc,axiom,
    ! [M8: set_nat,N5: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M8 @ N5 )
      = ( ( image_nat_nat @ suc @ M8 )
        = N5 ) ) ).

% bij_betw_Suc
thf(fact_9708_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J2: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J2 ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J2 ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_9709_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J2: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J2 ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J2 ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_9710_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_9711_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_9712_image__int__atLeastLessThan,axiom,
    ! [A: nat,B: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
      = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% image_int_atLeastLessThan
thf(fact_9713_int__in__range__abs,axiom,
    ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).

% int_in_range_abs
thf(fact_9714_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_9715_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_9716_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_9717_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_9718_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_9719_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_9720_range__mod,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( image_nat_nat
          @ ^ [M5: nat] : ( modulo_modulo_nat @ M5 @ N )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% range_mod
thf(fact_9721_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X2: int] : ( plus_plus_int @ X2 @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9722_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ X2 @ L )
        @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L ) ) )
      = ( set_or8404916559141939852nteger @ L @ U ) ) ).

% image_add_integer_atLeastLessThan
thf(fact_9723_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ U )
     => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
        = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_int
thf(fact_9724_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ 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
                @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9725_setceilmax,axiom,
    ! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ S @ M )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Listy ) )
           => ( vEBT_invar_vebt @ X3 @ N ) )
       => ( ( M
            = ( suc @ N ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Listy ) )
               => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ X3 ) )
                  = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
           => ( ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ S ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) )
             => ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ S @ ( set_VEBT_VEBT2 @ Listy ) ) ) ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ) ).

% setceilmax
thf(fact_9726_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ T ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).

% height_compose_list
thf(fact_9727_max__ins__scaled,axiom,
    ! [N: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( vEBT_VEBT_height @ X14 ) ) @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ ( lattic8265883725875713057ax_nat @ ( insert_nat @ ( vEBT_VEBT_height @ X14 ) @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ).

% max_ins_scaled
thf(fact_9728_height__i__max,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ X13 ) )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I ) ) @ ( ord_max_nat @ Foo @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ).

% height_i_max
thf(fact_9729_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,N: nat,X14: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ X13 ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ N @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I ) ) ) @ ( suc @ ( suc @ ( times_times_nat @ N @ ( ord_max_nat @ ( vEBT_VEBT_height @ X14 ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ) ) ).

% max_idx_list
thf(fact_9730_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D: nat] : ( dvd_dvd_nat @ D @ N ) ) )
        = N ) ) ).

% Max_divisors_self_nat
thf(fact_9731_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N4: nat] :
          ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N4 ) @ M5 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9732_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu2 @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).

% VEBT_internal.height.simps(2)
thf(fact_9733_VEBT__internal_Oheight_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_height @ X )
        = Y )
     => ( ( ? [A6: $o,B5: $o] :
              ( X
              = ( vEBT_Leaf @ A6 @ B5 ) )
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) )
             => ( Y
               != ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.height.elims
thf(fact_9734_VEBT__internal_Oheight_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_height @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ X )
       => ( ! [A6: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A6 @ B5 ) )
             => ( ( Y = zero_zero_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A6 @ B5 ) ) ) )
         => ~ ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList2 ) ) ) ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu @ Deg2 @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.height.pelims
thf(fact_9735_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_9736_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D: int] : ( dvd_dvd_int @ D @ N ) ) )
        = ( abs_abs_int @ N ) ) ) ).

% Max_divisors_self_int
thf(fact_9737_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).

% min_Suc_Suc
thf(fact_9738_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9739_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9740_min__minus_H,axiom,
    ! [M: nat,K: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ K ) @ M )
      = ( minus_minus_nat @ M @ K ) ) ).

% min_minus'
thf(fact_9741_min__minus,axiom,
    ! [M: nat,K: nat] :
      ( ( ord_min_nat @ M @ ( minus_minus_nat @ M @ K ) )
      = ( minus_minus_nat @ M @ K ) ) ).

% min_minus
thf(fact_9742_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ ( suc @ A ) @ B )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(1)
thf(fact_9743_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ B @ ( suc @ A ) )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(2)
thf(fact_9744_rev__min__pm1,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ ( ord_min_nat @ B @ A ) )
      = A ) ).

% rev_min_pm1
thf(fact_9745_rev__min__pm,axiom,
    ! [B: nat,A: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ B @ A ) @ ( minus_minus_nat @ A @ B ) )
      = A ) ).

% rev_min_pm
thf(fact_9746_min__pm1,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ ( ord_min_nat @ A @ B ) )
      = A ) ).

% min_pm1
thf(fact_9747_min__pm,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ A @ B ) @ ( minus_minus_nat @ A @ B ) )
      = A ) ).

% min_pm
thf(fact_9748_min__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).

% min_numeral_Suc
thf(fact_9749_min__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).

% min_Suc_numeral
thf(fact_9750_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).

% nat_mult_min_left
thf(fact_9751_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_min_right
thf(fact_9752_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).

% min_diff
thf(fact_9753_concat__bit__assoc__sym,axiom,
    ! [M: nat,N: nat,K: int,L: int,R2: int] :
      ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R2 )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L @ R2 ) ) ) ).

% concat_bit_assoc_sym
thf(fact_9754_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L ) )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L ) ) ) ).

% take_bit_concat_bit_eq
thf(fact_9755_mod__mod__power,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ K @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( modulo_modulo_nat @ K @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( ord_min_nat @ M @ N ) ) ) ) ).

% mod_mod_power
thf(fact_9756_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(3)
thf(fact_9757_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(2)
thf(fact_9758_shiftl__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_Sh3965577149348748681tl_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% shiftl_Suc_0
thf(fact_9759_shiftr__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_Sh2154871086232339855tr_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% shiftr_Suc_0
thf(fact_9760_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_9761_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_9762_inj__Suc,axiom,
    ! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).

% inj_Suc
thf(fact_9763_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( ( member_nat @ N2 @ N5 )
         => ( ord_less_eq_nat @ K @ N2 ) )
     => ( inj_on_nat_nat
        @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ K )
        @ N5 ) ) ).

% inj_on_diff_nat
thf(fact_9764_MVT2,axiom,
    ! [A: real,B: real,F: real > real,F5: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
       => ? [Z6: real] :
            ( ( ord_less_real @ A @ Z6 )
            & ( ord_less_real @ Z6 @ B )
            & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F5 @ Z6 ) ) ) ) ) ) ).

% MVT2
thf(fact_9765_DERIV__local__const,axiom,
    ! [F: real > real,L: real,X: real,D2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D2 )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D2 )
             => ( ( F @ X )
                = ( F @ Y3 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9766_DERIV__ln,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_ln
thf(fact_9767_DERIV__neg__dec__right,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 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( ord_less_real @ H5 @ D3 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H5 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9768_DERIV__pos__inc__right,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 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( ord_less_real @ H5 @ D3 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H5 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9769_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: real > real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
          = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).

% DERIV_const_ratio_const
thf(fact_9770_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9771_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9772_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_9773_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_9774_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: real > real,G2: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
         => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
       => ( ( ord_less_eq_real @ A @ B )
         => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_9775_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 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( ord_less_real @ H5 @ D3 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H5 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9776_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 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( ord_less_real @ H5 @ D3 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H5 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9777_DERIV__isconst__all,axiom,
    ! [F: real > real,X: real,Y: real] :
      ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( F @ X )
        = ( F @ Y ) ) ) ).

% DERIV_isconst_all
thf(fact_9778_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: real > real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
          = K ) ) ) ).

% DERIV_const_ratio_const2
thf(fact_9779_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L: real,X: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( member_real @ ( minus_minus_real @ X @ H5 ) @ S2 )
                 => ( ( ord_less_real @ H5 @ D3 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H5 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9780_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L: real,X: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( member_real @ ( minus_minus_real @ X @ H5 ) @ S2 )
                 => ( ( ord_less_real @ H5 @ D3 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H5 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9781_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L: real,X: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( member_real @ ( plus_plus_real @ X @ H5 ) @ S2 )
                 => ( ( ord_less_real @ H5 @ D3 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H5 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9782_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L: real,X: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S2 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H5: real] :
                ( ( ord_less_real @ zero_zero_real @ H5 )
               => ( ( member_real @ ( plus_plus_real @ X @ H5 ) @ S2 )
                 => ( ( ord_less_real @ H5 @ D3 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H5 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9783_DERIV__const__average,axiom,
    ! [A: real,B: real,V: real > real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ 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_9784_DERIV__local__min,axiom,
    ! [F: real > real,L: real,X: real,D2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D2 )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D2 )
             => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9785_DERIV__local__max,axiom,
    ! [F: real > real,L: real,X: real,D2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D2 )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D2 )
             => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9786_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_9787_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X2: real] : ( power_power_real @ X2 @ N )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S ) ) ).

% DERIV_pow
thf(fact_9788_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X: real,N: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X2: real] : ( power_power_real @ ( G @ X2 ) @ N )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9789_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z7: real] : ( powr_real @ Z7 @ 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_9790_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X: real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ R2 )
          @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9791_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_9792_DERIV__powr,axiom,
    ! [G: real > real,M: real,X: real,F: real > real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( 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
            @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) )
            @ ( 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 @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9793_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_9794_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_9795_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A2: 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 @ A2 ) ) ).

% arsinh_real_has_field_derivative
thf(fact_9796_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_9797_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: 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 @ A2 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_9798_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: 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 @ A2 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9799_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_9800_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_9801_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_9802_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M3: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ? [T4: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9803_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M3: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ? [T4: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
          & ( ( F @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9804_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_9805_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 )
         => ( ! [M3: nat,T4: real] :
                ( ( ( ord_less_nat @ M3 @ N )
                  & ( ord_less_eq_real @ H2 @ T4 )
                  & ( ord_less_eq_real @ T4 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
           => ? [T4: real] :
                ( ( ord_less_real @ H2 @ T4 )
                & ( ord_less_real @ T4 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9806_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 )
       => ( ! [M3: nat,T4: real] :
              ( ( ( ord_less_nat @ M3 @ N )
                & ( ord_less_eq_real @ zero_zero_real @ T4 )
                & ( ord_less_eq_real @ T4 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
         => ? [T4: real] :
              ( ( ord_less_real @ zero_zero_real @ T4 )
              & ( ord_less_eq_real @ T4 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                    @ ( set_ord_lessThan_nat @ N ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9807_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 )
         => ( ! [M3: nat,T4: real] :
                ( ( ( ord_less_nat @ M3 @ N )
                  & ( ord_less_eq_real @ zero_zero_real @ T4 )
                  & ( ord_less_eq_real @ T4 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
           => ? [T4: real] :
                ( ( ord_less_real @ zero_zero_real @ T4 )
                & ( ord_less_real @ T4 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9808_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 )
         => ( ! [M3: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
           => ? [T4: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T4 ) )
                & ( ord_less_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
                & ( ( F @ X )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9809_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M3: nat,T4: real] :
            ( ( ( ord_less_nat @ M3 @ N )
              & ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
       => ? [T4: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9810_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 )
       => ( ! [M3: nat,T4: real] :
              ( ( ( ord_less_nat @ M3 @ N )
                & ( ord_less_eq_real @ A @ T4 )
                & ( ord_less_eq_real @ T4 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ? [T4: real] :
                  ( ( ord_less_real @ A @ T4 )
                  & ( ord_less_real @ T4 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M5 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9811_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 )
       => ( ! [M3: nat,T4: real] :
              ( ( ( ord_less_nat @ M3 @ N )
                & ( ord_less_eq_real @ A @ T4 )
                & ( ord_less_eq_real @ T4 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B )
             => ? [T4: real] :
                  ( ( ord_less_real @ C @ T4 )
                  & ( ord_less_real @ T4 @ B )
                  & ( ( F @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M5 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9812_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 )
       => ( ! [M3: nat,T4: real] :
              ( ( ( ord_less_nat @ M3 @ N )
                & ( ord_less_eq_real @ A @ T4 )
                & ( ord_less_eq_real @ T4 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ 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 )
                   => ? [T4: real] :
                        ( ( ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ X @ T4 )
                            & ( ord_less_real @ T4 @ C ) ) )
                        & ( ~ ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ C @ T4 )
                            & ( ord_less_real @ T4 @ X ) ) )
                        & ( ( F @ X )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M5 ) )
                              @ ( set_ord_lessThan_nat @ N ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9813_inj__sgn__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( inj_on_real_real
        @ ^ [Y5: real] : ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_9814_Maclaurin__lemma2,axiom,
    ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B3: real] :
      ( ! [M3: nat,T4: real] :
          ( ( ( ord_less_nat @ M3 @ N )
            & ( ord_less_eq_real @ zero_zero_real @ T4 )
            & ( ord_less_eq_real @ T4 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
     => ( ( N
          = ( suc @ K ) )
       => ! [M2: nat,T7: real] :
            ( ( ( ord_less_nat @ M2 @ N )
              & ( ord_less_eq_real @ zero_zero_real @ T7 )
              & ( ord_less_eq_real @ T7 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M2 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P3 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P3 ) ) @ ( power_power_real @ U2 @ P3 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
                    @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T7 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P3 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P3 ) ) @ ( power_power_real @ T7 @ P3 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
                  @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9815_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_9816_DERIV__power__series_H,axiom,
    ! [R3: real,F: nat > real,X0: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R3 ) @ R3 ) )
         => ( summable_real
            @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X3 @ N4 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R3 ) @ R3 ) )
       => ( ( ord_less_real @ zero_zero_real @ R3 )
         => ( has_fi5821293074295781190e_real
            @ ^ [X2: real] :
                ( suminf_real
                @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ X2 @ ( suc @ N4 ) ) ) )
            @ ( suminf_real
              @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X0 @ N4 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9817_DERIV__isconst3,axiom,
    ! [A: real,B: real,X: real,Y: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
       => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
           => ( ( F @ X )
              = ( F @ Y ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_9818_DERIV__series_H,axiom,
    ! [F: real > nat > real,F5: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
      ( ! [N2: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X2: real] : ( F @ X2 @ N2 )
          @ ( F5 @ X0 @ N2 )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
           => ( summable_real @ ( F @ X3 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ( summable_real @ ( F5 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N2: nat,X3: real,Y3: real] :
                    ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
                   => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N2 ) @ ( F @ Y3 @ N2 ) ) ) @ ( times_times_real @ ( L5 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X2: real] : ( suminf_real @ ( F @ X2 ) )
                  @ ( suminf_real @ ( F5 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_9819_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_9820_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 )
       => ? [R: real] :
            ( ( ord_less_real @ zero_zero_real @ R )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X4 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_9821_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 )
       => ? [R: real] :
            ( ( ord_less_real @ zero_zero_real @ R )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R ) )
               => ( ( F @ X4 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_9822_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 )
       => ? [R: real] :
            ( ( ord_less_real @ zero_zero_real @ R )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R ) )
               => ( ord_less_real @ ( F @ X4 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_9823_isCont__real__sqrt,axiom,
    ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).

% isCont_real_sqrt
thf(fact_9824_isCont__real__root,axiom,
    ! [X: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N ) ) ).

% isCont_real_root
thf(fact_9825_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_9826_isCont__inverse__function2,axiom,
    ! [A: real,X: real,B: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ( ! [Z6: real] :
              ( ( ord_less_eq_real @ A @ Z6 )
             => ( ( ord_less_eq_real @ Z6 @ B )
               => ( ( G @ ( F @ Z6 ) )
                  = Z6 ) ) )
         => ( ! [Z6: real] :
                ( ( ord_less_eq_real @ A @ Z6 )
               => ( ( ord_less_eq_real @ Z6 @ B )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z6 @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_9827_isCont__ln,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ln_ln_real ) ) ).

% isCont_ln
thf(fact_9828_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
      = ( set_or4266950643985792945nteger @ L @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_9829_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_9830_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X2: real] : ( divide_divide_real @ ( cos_real @ X2 ) @ ( sin_real @ X2 ) )
    @ ( 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_9831_DERIV__inverse__function,axiom,
    ! [F: real > real,D4: real,G: real > real,X: real,A: real,B: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
     => ( ( D4 != zero_zero_real )
       => ( ( ord_less_real @ A @ X )
         => ( ( ord_less_real @ X @ B )
           => ( ! [Y3: real] :
                  ( ( ord_less_real @ A @ Y3 )
                 => ( ( ord_less_real @ Y3 @ B )
                   => ( ( F @ ( G @ Y3 ) )
                      = Y3 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
               => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_9832_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_9833_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_9834_LIM__less__bound,axiom,
    ! [B: real,X: real,F: real > real] :
      ( ( ord_less_real @ B @ X )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B @ X ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).

% LIM_less_bound
thf(fact_9835_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_9836_isCont__inverse__function,axiom,
    ! [D2: real,X: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D2 )
     => ( ! [Z6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z6 @ X ) ) @ D2 )
           => ( ( G @ ( F @ Z6 ) )
              = Z6 ) )
       => ( ! [Z6: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z6 @ X ) ) @ D2 )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z6 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_9837_GMVT_H,axiom,
    ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F5: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [Z6: real] :
            ( ( ord_less_eq_real @ A @ Z6 )
           => ( ( ord_less_eq_real @ Z6 @ B )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z6 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z6: real] :
              ( ( ord_less_eq_real @ A @ Z6 )
             => ( ( ord_less_eq_real @ Z6 @ B )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z6 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z6: real] :
                ( ( ord_less_real @ A @ Z6 )
               => ( ( ord_less_real @ Z6 @ B )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z6 ) @ ( topolo2177554685111907308n_real @ Z6 @ top_top_set_real ) ) ) )
           => ( ! [Z6: real] :
                  ( ( ord_less_real @ A @ Z6 )
                 => ( ( ord_less_real @ Z6 @ B )
                   => ( has_fi5821293074295781190e_real @ F @ ( F5 @ Z6 ) @ ( topolo2177554685111907308n_real @ Z6 @ 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 ) ) @ ( F5 @ C2 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_9838_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 )
         => ! [N10: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_9839_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 ) )
         => ! [N10: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_9840_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_9841_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X2: nat] : ( times_times_nat @ X2 @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_9842_LIMSEQ__root,axiom,
    ( filterlim_nat_real
    @ ^ [N4: nat] : ( root @ N4 @ ( semiri5074537144036343181t_real @ N4 ) )
    @ ( topolo2815343760600316023s_real @ one_one_real )
    @ at_top_nat ) ).

% LIMSEQ_root
thf(fact_9843_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
              @ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L4: real] :
                ( ! [N10: nat] : ( ord_less_eq_real @ ( F @ N10 ) @ L4 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
                & ! [N10: nat] : ( ord_less_eq_real @ L4 @ ( G @ N10 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_9844_LIMSEQ__inverse__zero,axiom,
    ! [X9: nat > real] :
      ( ! [R: real] :
        ? [N11: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N11 @ N2 )
         => ( ord_less_real @ R @ ( X9 @ N2 ) ) )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( inverse_inverse_real @ ( X9 @ N4 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_9845_lim__inverse__n_H,axiom,
    ( filterlim_nat_real
    @ ^ [N4: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N4 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% lim_inverse_n'
thf(fact_9846_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N4: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_9847_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( root @ N4 @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_9848_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N4: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_9849_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 )
             => ? [N10: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N10 ) @ E2 ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_9850_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_9851_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N4 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_9852_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_9853_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_9854_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N4 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_9855_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N4: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9856_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] :
      ( filterlim_nat_real
      @ ^ [N4: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ) @ N4 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ at_top_nat ) ).

% tendsto_exp_limit_sequentially
thf(fact_9857_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N4: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9858_summable__Leibniz_I1_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( summable_real
          @ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_9859_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
            @ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ) ).

% summable
thf(fact_9860_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 )
     => ~ ! [K2: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_9861_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K2: nat > int] :
          ( filterlim_nat_real
          @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_9862_summable__Leibniz_I4_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N4: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_9863_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_9864_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
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( suminf_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_9865_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N4: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_9866_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] :
              ( ! [N10: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N4: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N10: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N4: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_9867_summable__Leibniz_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N4: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_9868_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
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_9869_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N4: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_9870_real__bounded__linear,axiom,
    ( real_V5970128139526366754l_real
    = ( ^ [F6: real > real] :
        ? [C4: real] :
          ( F6
          = ( ^ [X2: real] : ( times_times_real @ X2 @ C4 ) ) ) ) ) ).

% real_bounded_linear
thf(fact_9871_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_9872_tendsto__exp__limit__at__right,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y5: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y5 ) ) @ ( divide_divide_real @ one_one_real @ Y5 ) )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% tendsto_exp_limit_at_right
thf(fact_9873_tendsto__arctan__at__bot,axiom,
    filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).

% tendsto_arctan_at_bot
thf(fact_9874_ln__at__0,axiom,
    filterlim_real_real @ ln_ln_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).

% ln_at_0
thf(fact_9875_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_9876_exp__at__bot,axiom,
    filterlim_real_real @ exp_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_bot_real ).

% exp_at_bot
thf(fact_9877_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).

% filterlim_inverse_at_bot_neg
thf(fact_9878_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_9879_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_9880_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ X3 @ B )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y4 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_9881_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: real > real,F2: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F2 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N )
            @ at_bot_real
            @ F2 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_9882_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: real > real,F2: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F2 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N )
            @ at_top_real
            @ F2 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_9883_sqrt__at__top,axiom,
    filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).

% sqrt_at_top
thf(fact_9884_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_9885_greaterThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).

% greaterThan_Suc
thf(fact_9886_ln__x__over__x__tendsto__0,axiom,
    ( filterlim_real_real
    @ ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_real ) ).

% ln_x_over_x_tendsto_0
thf(fact_9887_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).

% filterlim_inverse_at_right_top
thf(fact_9888_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).

% filterlim_inverse_at_top_right
thf(fact_9889_tendsto__power__div__exp__0,axiom,
    ! [K: nat] :
      ( filterlim_real_real
      @ ^ [X2: real] : ( divide_divide_real @ ( power_power_real @ X2 @ K ) @ ( exp_real @ X2 ) )
      @ ( topolo2815343760600316023s_real @ zero_zero_real )
      @ at_top_real ) ).

% tendsto_power_div_exp_0
thf(fact_9890_tendsto__exp__limit__at__top,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y5: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y5 ) ) @ Y5 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ at_top_real ) ).

% tendsto_exp_limit_at_top
thf(fact_9891_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_9892_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_9893_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ B @ X3 )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ Y4 @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_neg_imp_decreasing_at_top
thf(fact_9894_lhopital__left__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top
thf(fact_9895_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I2: nat] : ( P @ ( suc @ I2 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_9896_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N4: nat] : ( P @ ( plus_plus_nat @ N4 @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_9897_sequentially__offset,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I2: nat] : ( P @ ( plus_plus_nat @ I2 @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_9898_eventually__at__right__to__0,axiom,
    ! [P: real > $o,A: real] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
      = ( eventually_real
        @ ^ [X2: real] : ( P @ ( plus_plus_real @ X2 @ A ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_right_to_0
thf(fact_9899_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( eventually_real
        @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ A @ B ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).

% eventually_at_right_real
thf(fact_9900_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( eventually_real
        @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ B @ A ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).

% eventually_at_left_real
thf(fact_9901_eventually__at__right__to__top,axiom,
    ! [P: real > $o] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
      = ( eventually_real
        @ ^ [X2: real] : ( P @ ( inverse_inverse_real @ X2 ) )
        @ at_top_real ) ) ).

% eventually_at_right_to_top
thf(fact_9902_eventually__at__top__to__right,axiom,
    ! [P: real > $o] :
      ( ( eventually_real @ P @ at_top_real )
      = ( eventually_real
        @ ^ [X2: real] : ( P @ ( inverse_inverse_real @ X2 ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_top_to_right
thf(fact_9903_lhopital__at__top__at__top,axiom,
    ! [F: real > real,A: real,G: real > real,F5: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_top
thf(fact_9904_lhopital,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F2: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                    @ F2
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                    @ F2
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).

% lhopital
thf(fact_9905_lhopital__right__at__top__at__top,axiom,
    ! [F: real > real,A: real,G: real > real,F5: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
thf(fact_9906_lhopital__at__top__at__bot,axiom,
    ! [F: real > real,A: real,G: real > real,F5: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_bot
thf(fact_9907_lhopital__left__at__top__at__top,axiom,
    ! [F: real > real,A: real,G: real > real,F5: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
thf(fact_9908_lhopital__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top
thf(fact_9909_lhospital__at__top__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F5: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ at_top_real )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ at_top_real )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ at_top_real )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real ) ) ) ) ) ) ).

% lhospital_at_top_at_top
thf(fact_9910_lhopital__right,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F2: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                    @ F2
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                    @ F2
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right
thf(fact_9911_lhopital__right__0,axiom,
    ! [F0: real > real,G0: real > real,G2: real > real,F5: real > real,F2: filter_real] :
      ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G0 @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                    @ F2
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F0 @ X2 ) @ ( G0 @ X2 ) )
                    @ F2
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right_0
thf(fact_9912_lhopital__left,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F2: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                    @ F2
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                    @ F2
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_left
thf(fact_9913_lhopital__right__at__top__at__bot,axiom,
    ! [F: real > real,A: real,G: real > real,F5: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
     => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
thf(fact_9914_lhopital__left__at__top__at__bot,axiom,
    ! [F: real > real,A: real,G: real > real,F5: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
     => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
thf(fact_9915_lhopital__right__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top
thf(fact_9916_lhopital__right__0__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F5: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F5 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).

% lhopital_right_0_at_top
thf(fact_9917_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_9918_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_9919_rat__inverse__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A4 ) @ B2 ) @ ( abs_abs_int @ A4 ) ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_inverse_code
thf(fact_9920_quotient__of__number_I3_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).

% quotient_of_number(3)
thf(fact_9921_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9922_quotient__of__number_I5_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).

% quotient_of_number(5)
thf(fact_9923_quotient__of__div,axiom,
    ! [R2: rat,N: int,D2: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ N @ D2 ) )
     => ( R2
        = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D2 ) ) ) ) ).

% quotient_of_div
thf(fact_9924_divide__rat__def,axiom,
    ( divide_divide_rat
    = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).

% divide_rat_def
thf(fact_9925_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_9926_quotient__of__denom__pos,axiom,
    ! [R2: rat,P4: int,Q2: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P4 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% quotient_of_denom_pos
thf(fact_9927_rat__floor__code,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [P3: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P3 ) ) ) ) ).

% rat_floor_code
thf(fact_9928_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
      = ( set_or2715278749043346189nteger @ L @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_9929_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P3: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A4: int,C4: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B2: int,D: int] : ( ord_less_int @ ( times_times_int @ A4 @ D ) @ ( times_times_int @ C4 @ B2 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P3 ) ) ) ) ).

% rat_less_code
thf(fact_9930_rat__plus__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A4 @ D ) @ ( times_times_int @ B2 @ C4 ) ) @ ( times_times_int @ C4 @ D ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_plus_code
thf(fact_9931_GMVT,axiom,
    ! [A: real,B: real,F: real > real,G: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ( ord_less_eq_real @ A @ X3 )
              & ( ord_less_eq_real @ X3 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
       => ( ! [X3: real] :
              ( ( ( ord_less_real @ A @ X3 )
                & ( ord_less_real @ X3 @ B ) )
             => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
         => ( ! [X3: real] :
                ( ( ( ord_less_eq_real @ A @ X3 )
                  & ( ord_less_eq_real @ X3 @ B ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G ) )
           => ( ! [X3: real] :
                  ( ( ( ord_less_real @ A @ X3 )
                    & ( ord_less_real @ X3 @ B ) )
                 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X3 @ 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_9932_normalize__denom__zero,axiom,
    ! [P4: int] :
      ( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_9933_normalize__negative,axiom,
    ! [Q2: int,P4: int] :
      ( ( ord_less_int @ Q2 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).

% normalize_negative
thf(fact_9934_diff__rat__def,axiom,
    ( minus_minus_rat
    = ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).

% diff_rat_def
thf(fact_9935_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R2 )
     => ~ ! [S3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S3 )
           => ! [T4: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T4 )
               => ( R2
                 != ( plus_plus_rat @ S3 @ T4 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9936_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P4: int,Q2: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P4 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% normalize_denom_pos
thf(fact_9937_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P4: int,R2: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
            = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
         => ( ( times_times_int @ P4 @ S )
            = ( times_times_int @ R2 @ Q2 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9938_rat__divide__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ D ) @ ( times_times_int @ C4 @ B2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_divide_code
thf(fact_9939_Frct__code__post_I5_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% Frct_code_post(5)
thf(fact_9940_MVT,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B )
               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ? [L4: real,Z6: real] :
              ( ( ord_less_real @ A @ Z6 )
              & ( ord_less_real @ Z6 @ B )
              & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z6 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).

% MVT
thf(fact_9941_Rolle__deriv,axiom,
    ! [A: real,B: real,F: real > real,F5: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ A )
          = ( F @ B ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ! [X3: real] :
                ( ( ord_less_real @ A @ X3 )
               => ( ( ord_less_real @ X3 @ B )
                 => ( has_de1759254742604945161l_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
           => ? [Z6: real] :
                ( ( ord_less_real @ A @ Z6 )
                & ( ord_less_real @ Z6 @ B )
                & ( ( F5 @ Z6 )
                  = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_9942_mvt,axiom,
    ! [A: real,B: real,F: real > real,F5: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B )
               => ( has_de1759254742604945161l_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ~ ! [Xi3: real] :
                ( ( ord_less_real @ A @ Xi3 )
               => ( ( ord_less_real @ Xi3 @ B )
                 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                   != ( F5 @ Xi3 @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).

% mvt
thf(fact_9943_Frct__code__post_I2_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_9944_Frct__code__post_I1_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_9945_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ( ( F @ B )
            = ( F @ A ) ) ) ) ) ).

% DERIV_isconst_end
thf(fact_9946_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_real @ A @ X3 )
           => ( ( ord_less_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).

% DERIV_neg_imp_decreasing_open
thf(fact_9947_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_real @ A @ X3 )
           => ( ( ord_less_real @ X3 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_9948_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: real > real,X: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ( ( ord_less_eq_real @ A @ X )
           => ( ( ord_less_eq_real @ X @ B )
             => ( ( F @ X )
                = ( F @ A ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_9949_Rolle,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ A )
          = ( F @ B ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ! [X3: real] :
                ( ( ord_less_real @ A @ X3 )
               => ( ( ord_less_real @ X3 @ B )
                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
           => ? [Z6: real] :
                ( ( ord_less_real @ A @ Z6 )
                & ( ord_less_real @ Z6 @ B )
                & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z6 @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_9950_Frct__code__post_I4_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
      = ( numeral_numeral_rat @ K ) ) ).

% Frct_code_post(4)
thf(fact_9951_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
      = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).

% Frct_code_post(6)
thf(fact_9952_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J2: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J2 @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J2 ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9953_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_9954_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_9955_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I ) @ J2 )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J2 ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J2 ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_9956_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ J2 )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J2 ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J2 ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_9957_atLeast__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).

% atLeast_Suc
thf(fact_9958_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J2: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J2 @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J2 ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9959_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
              @ ^ [L3: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_9960_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_9961_zero__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_int ) ).

% zero_integer.rep_eq
thf(fact_9962_int__of__integer__numeral,axiom,
    ! [K: num] :
      ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% int_of_integer_numeral
thf(fact_9963_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa3 ) )
      = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).

% plus_integer.rep_eq
thf(fact_9964_complex__Re__numeral,axiom,
    ! [V: num] :
      ( ( re @ ( numera6690914467698888265omplex @ V ) )
      = ( numeral_numeral_real @ V ) ) ).

% complex_Re_numeral
thf(fact_9965_divide__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa3 ) )
      = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).

% divide_integer.rep_eq
thf(fact_9966_Re__divide__of__nat,axiom,
    ! [Z: complex,N: nat] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% Re_divide_of_nat
thf(fact_9967_Re__divide__of__real,axiom,
    ! [Z: complex,R2: real] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ R2 ) ) ).

% Re_divide_of_real
thf(fact_9968_Re__sgn,axiom,
    ! [Z: complex] :
      ( ( re @ ( sgn_sgn_complex @ Z ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% Re_sgn
thf(fact_9969_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).

% Re_divide_numeral
thf(fact_9970_cos__Arg__i__mult__zero,axiom,
    ! [Y: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( ( re @ Y )
          = zero_zero_real )
       => ( ( cos_real @ ( arg @ Y ) )
          = zero_zero_real ) ) ) ).

% cos_Arg_i_mult_zero
thf(fact_9971_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_9972_plus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).

% plus_complex.simps(1)
thf(fact_9973_dup_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( code_dup @ X ) )
      = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).

% dup.rep_eq
thf(fact_9974_imaginary__unit_Osimps_I1_J,axiom,
    ( ( re @ imaginary_unit )
    = zero_zero_real ) ).

% imaginary_unit.simps(1)
thf(fact_9975_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
      = ( ord_le6747313008572928689nteger @ X @ Y ) ) ).

% int_of_integer_less_iff
thf(fact_9976_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_9977_integer__less__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [K3: code_integer,L3: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).

% integer_less_iff
thf(fact_9978_zero__complex_Osimps_I1_J,axiom,
    ( ( re @ zero_zero_complex )
    = zero_zero_real ) ).

% zero_complex.simps(1)
thf(fact_9979_unset__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa3: code_integer] :
      ( ( code_int_of_integer @ ( bit_se8260200283734997820nteger @ X @ Xa3 ) )
      = ( bit_se4203085406695923979it_int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).

% unset_bit_integer.rep_eq
thf(fact_9980_Re__csqrt,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).

% Re_csqrt
thf(fact_9981_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
      = ( ( re @ Z )
        = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% cmod_plus_Re_le_0_iff
thf(fact_9982_bin__last__integer_Orep__eq,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [X2: code_integer] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ X2 ) ) ) ) ).

% bin_last_integer.rep_eq
thf(fact_9983_bin__rest__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( bits_b2549910563261871055nteger @ X ) )
      = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% bin_rest_integer.rep_eq
thf(fact_9984_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_9985_Bit__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa3: $o] :
      ( ( code_int_of_integer @ ( bits_Bit_integer @ X @ Xa3 ) )
      = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ Xa3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ X ) ) ) ) ).

% Bit_integer.rep_eq
thf(fact_9986_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L3: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L3 ) @ ( modulo364778990260209775nteger @ K3 @ L3 ) ) ) ) ).

% divmod_integer_def
thf(fact_9987_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L3: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_9988_csqrt_Ocode,axiom,
    ( csqrt
    = ( ^ [Z7: complex] :
          ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z7 ) @ ( re @ Z7 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          @ ( times_times_real
            @ ( if_real
              @ ( ( im @ Z7 )
                = zero_zero_real )
              @ one_one_real
              @ ( sgn_sgn_real @ ( im @ Z7 ) ) )
            @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z7 ) @ ( re @ Z7 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% csqrt.code
thf(fact_9989_complex__Im__fact,axiom,
    ! [N: nat] :
      ( ( im @ ( semiri5044797733671781792omplex @ N ) )
      = zero_zero_real ) ).

% complex_Im_fact
thf(fact_9990_complex__Im__of__int,axiom,
    ! [Z: int] :
      ( ( im @ ( ring_17405671764205052669omplex @ Z ) )
      = zero_zero_real ) ).

% complex_Im_of_int
thf(fact_9991_Im__complex__of__real,axiom,
    ! [Z: real] :
      ( ( im @ ( real_V4546457046886955230omplex @ Z ) )
      = zero_zero_real ) ).

% Im_complex_of_real
thf(fact_9992_Im__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( im @ ( power_power_complex @ X @ N ) )
        = zero_zero_real ) ) ).

% Im_power_real
thf(fact_9993_complex__Im__numeral,axiom,
    ! [V: num] :
      ( ( im @ ( numera6690914467698888265omplex @ V ) )
      = zero_zero_real ) ).

% complex_Im_numeral
thf(fact_9994_complex__Im__of__nat,axiom,
    ! [N: nat] :
      ( ( im @ ( semiri8010041392384452111omplex @ N ) )
      = zero_zero_real ) ).

% complex_Im_of_nat
thf(fact_9995_Im__divide__of__real,axiom,
    ! [Z: complex,R2: real] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R2 ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ R2 ) ) ).

% Im_divide_of_real
thf(fact_9996_Im__sgn,axiom,
    ! [Z: complex] :
      ( ( im @ ( sgn_sgn_complex @ Z ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% Im_sgn
thf(fact_9997_Re__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( re @ ( power_power_complex @ X @ N ) )
        = ( power_power_real @ ( re @ X ) @ N ) ) ) ).

% Re_power_real
thf(fact_9998_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).

% Im_divide_numeral
thf(fact_9999_Im__divide__of__nat,axiom,
    ! [Z: complex,N: nat] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% Im_divide_of_nat
thf(fact_10000_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
       => ( ( csqrt @ X )
          = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).

% csqrt_of_real_nonneg
thf(fact_10001_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_10002_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_10003_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( ( member_complex @ Z @ ring_1_Ints_complex )
      = ( ( ( im @ Z )
          = zero_zero_real )
        & ? [I2: int] :
            ( ( re @ Z )
            = ( ring_1_of_int_real @ I2 ) ) ) ) ).

% complex_is_Int_iff
thf(fact_10004_one__complex_Osimps_I2_J,axiom,
    ( ( im @ one_one_complex )
    = zero_zero_real ) ).

% one_complex.simps(2)
thf(fact_10005_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_10006_zero__complex_Osimps_I2_J,axiom,
    ( ( im @ zero_zero_complex )
    = zero_zero_real ) ).

% zero_complex.simps(2)
thf(fact_10007_plus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).

% plus_complex.simps(2)
thf(fact_10008_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_10009_cmod__eq__Re,axiom,
    ! [Z: complex] :
      ( ( ( im @ Z )
        = zero_zero_real )
     => ( ( real_V1022390504157884413omplex @ Z )
        = ( abs_abs_real @ ( re @ Z ) ) ) ) ).

% cmod_eq_Re
thf(fact_10010_cmod__eq__Im,axiom,
    ! [Z: complex] :
      ( ( ( re @ Z )
        = zero_zero_real )
     => ( ( real_V1022390504157884413omplex @ Z )
        = ( abs_abs_real @ ( im @ Z ) ) ) ) ).

% cmod_eq_Im
thf(fact_10011_Im__eq__0,axiom,
    ! [Z: complex] :
      ( ( ( abs_abs_real @ ( re @ Z ) )
        = ( real_V1022390504157884413omplex @ Z ) )
     => ( ( im @ Z )
        = zero_zero_real ) ) ).

% Im_eq_0
thf(fact_10012_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_10013_plus__complex_Ocode,axiom,
    ( plus_plus_complex
    = ( ^ [X2: complex,Y5: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X2 ) @ ( re @ Y5 ) ) @ ( plus_plus_real @ ( im @ X2 ) @ ( im @ Y5 ) ) ) ) ) ).

% plus_complex.code
thf(fact_10014_scaleR__complex_Ocode,axiom,
    ( real_V2046097035970521341omplex
    = ( ^ [R5: real,X2: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X2 ) ) @ ( times_times_real @ R5 @ ( im @ X2 ) ) ) ) ) ).

% scaleR_complex.code
thf(fact_10015_csqrt__principal,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
      | ( ( ( re @ ( csqrt @ Z ) )
          = zero_zero_real )
        & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).

% csqrt_principal
thf(fact_10016_cmod__le,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).

% cmod_le
thf(fact_10017_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_10018_Re__exp,axiom,
    ! [Z: complex] :
      ( ( re @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).

% Re_exp
thf(fact_10019_Im__exp,axiom,
    ! [Z: complex] :
      ( ( im @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).

% Im_exp
thf(fact_10020_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_10021_times__complex_Ocode,axiom,
    ( times_times_complex
    = ( ^ [X2: complex,Y5: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y5 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y5 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y5 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y5 ) ) ) ) ) ) ).

% times_complex.code
thf(fact_10022_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_10023_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_10024_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_10025_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_10026_norm__complex__def,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z7: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z7 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z7 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% norm_complex_def
thf(fact_10027_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_10028_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_10029_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_10030_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_10031_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_10032_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_10033_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_10034_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_10035_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_10036_inverse__complex_Ocode,axiom,
    ( invers8013647133539491842omplex
    = ( ^ [X2: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X2 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X2 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% inverse_complex.code
thf(fact_10037_Complex__divide,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X2: complex,Y5: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y5 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y5 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y5 ) ) @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y5 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% Complex_divide
thf(fact_10038_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_10039_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_10040_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L3: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_10041_nat__of__integer__numeral,axiom,
    ! [N: num] :
      ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% nat_of_integer_numeral
thf(fact_10042_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] :
      ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_of_integer_code_post(3)
thf(fact_10043_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
     => ( ( code_nat_of_integer @ K )
        = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
thf(fact_10044_Re__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ Z @ R2 ) )
        = ( divide_divide_real @ ( re @ Z ) @ ( re @ R2 ) ) ) ) ).

% Re_divide_Reals
thf(fact_10045_Im__divide__Reals,axiom,
    ! [R2: complex,Z: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ Z @ R2 ) )
        = ( divide_divide_real @ ( im @ Z ) @ ( re @ R2 ) ) ) ) ).

% Im_divide_Reals
thf(fact_10046_complex__is__Real__iff,axiom,
    ! [Z: complex] :
      ( ( member_complex @ Z @ real_V2521375963428798218omplex )
      = ( ( im @ Z )
        = zero_zero_real ) ) ).

% complex_is_Real_iff
thf(fact_10047_Complex__in__Reals,axiom,
    ! [X: real] : ( member_complex @ ( complex2 @ X @ zero_zero_real ) @ real_V2521375963428798218omplex ) ).

% Complex_in_Reals
thf(fact_10048_nat__of__integer__code__post_I1_J,axiom,
    ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_nat ) ).

% nat_of_integer_code_post(1)
thf(fact_10049_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ord_less_nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
          = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ) ).

% nat_of_integer_less_iff
thf(fact_10050_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_10051_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_10052_card__Collect__less__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
      = N ) ).

% card_Collect_less_nat
thf(fact_10053_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_10054_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_atLeastLessThan
thf(fact_10055_complex__cnj__divide,axiom,
    ! [X: complex,Y: complex] :
      ( ( cnj @ ( divide1717551699836669952omplex @ X @ Y ) )
      = ( divide1717551699836669952omplex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).

% complex_cnj_divide
thf(fact_10056_card__Collect__le__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
      = ( suc @ N ) ) ).

% card_Collect_le_nat
thf(fact_10057_complex__cnj__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( cnj @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).

% complex_cnj_add
thf(fact_10058_card__UNIV__bool,axiom,
    ( ( finite_card_o @ top_top_set_o )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% card_UNIV_bool
thf(fact_10059_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_10060_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_10061_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_10062_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_10063_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_10064_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_10065_card__less__Suc2,axiom,
    ! [M8: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M8 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M8 )
                & ( ord_less_nat @ K3 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M8 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_10066_card__less__Suc,axiom,
    ! [M8: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M8 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M8 )
                  & ( ord_less_nat @ K3 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M8 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_10067_card__less,axiom,
    ! [M8: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M8 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M8 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_10068_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
     => ( A2
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_10069_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_10070_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_10071_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_10072_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_10073_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ S2 ) ) ).

% card_sum_le_nat_sum
thf(fact_10074_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z7: complex] :
                  ( ( power_power_complex @ Z7 @ N )
                  = C ) ) )
          = N ) ) ) ).

% card_nth_roots
thf(fact_10075_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z7: complex] :
                ( ( power_power_complex @ Z7 @ N )
                = one_one_complex ) ) )
        = N ) ) ).

% card_roots_unity_eq
thf(fact_10076_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_10077_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_10078_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_10079_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_10080_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_10081_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_10082_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_10083_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_10084_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_10085_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_10086_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_10087_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_10088_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_10089_complex__div__cnj,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A4: complex,B2: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A4 @ ( cnj @ B2 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_div_cnj
thf(fact_10090_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_10091_card__num0,axiom,
    ( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
    = zero_zero_nat ) ).

% card_num0
thf(fact_10092_card__nat,axiom,
    ( ( finite_card_nat @ top_top_set_nat )
    = zero_zero_nat ) ).

% card_nat
thf(fact_10093_card__literal,axiom,
    ( ( finite_card_literal @ top_top_set_literal )
    = zero_zero_nat ) ).

% card_literal
thf(fact_10094_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_10095_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_10096_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_10097_char_Osize_I2_J,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_10098_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_10099_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_10100_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_10101_char_Osize__gen,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size_gen
thf(fact_10102_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_10103_less__eq__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C6: $o,C7: $o] :
      ( ( ord_less_eq_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C6 @ C7 ) )
      = ( ord_less_eq_nat
        @ ( foldr_o_nat
          @ ^ [B2: $o,K3: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ B0 @ ( cons_o @ B1 @ ( cons_o @ B22 @ ( cons_o @ B32 @ ( cons_o @ B42 @ ( cons_o @ B52 @ ( cons_o @ B62 @ ( cons_o @ B72 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat )
        @ ( foldr_o_nat
          @ ^ [B2: $o,K3: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ C0 @ ( cons_o @ C1 @ ( cons_o @ C22 @ ( cons_o @ C32 @ ( cons_o @ C42 @ ( cons_o @ C52 @ ( cons_o @ C6 @ ( cons_o @ C7 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat ) ) ) ).

% less_eq_char_simp
thf(fact_10104_less__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C6: $o,C7: $o] :
      ( ( ord_less_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C52 @ C6 @ C7 ) )
      = ( ord_less_nat
        @ ( foldr_o_nat
          @ ^ [B2: $o,K3: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ B0 @ ( cons_o @ B1 @ ( cons_o @ B22 @ ( cons_o @ B32 @ ( cons_o @ B42 @ ( cons_o @ B52 @ ( cons_o @ B62 @ ( cons_o @ B72 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat )
        @ ( foldr_o_nat
          @ ^ [B2: $o,K3: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ C0 @ ( cons_o @ C1 @ ( cons_o @ C22 @ ( cons_o @ C32 @ ( cons_o @ C42 @ ( cons_o @ C52 @ ( cons_o @ C6 @ ( cons_o @ C7 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat ) ) ) ).

% less_char_simp
thf(fact_10105_less__char__def,axiom,
    ( ord_less_char
    = ( ^ [C12: char,C23: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C12 ) @ ( comm_s629917340098488124ar_nat @ C23 ) ) ) ) ).

% less_char_def
thf(fact_10106_upto_Opsimps,axiom,
    ! [I: int,J2: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J2 ) )
     => ( ( ( ord_less_eq_int @ I @ J2 )
         => ( ( upto @ I @ J2 )
            = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ I @ J2 )
         => ( ( upto @ I @ J2 )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_10107_upto__Nil,axiom,
    ! [I: int,J2: int] :
      ( ( ( upto @ I @ J2 )
        = nil_int )
      = ( ord_less_int @ J2 @ I ) ) ).

% upto_Nil
thf(fact_10108_upto__Nil2,axiom,
    ! [I: int,J2: int] :
      ( ( nil_int
        = ( upto @ I @ J2 ) )
      = ( ord_less_int @ J2 @ I ) ) ).

% upto_Nil2
thf(fact_10109_upto__empty,axiom,
    ! [J2: int,I: int] :
      ( ( ord_less_int @ J2 @ I )
     => ( ( upto @ I @ J2 )
        = nil_int ) ) ).

% upto_empty
thf(fact_10110_upto__single,axiom,
    ! [I: int] :
      ( ( upto @ I @ I )
      = ( cons_int @ I @ nil_int ) ) ).

% upto_single
thf(fact_10111_nth__upto,axiom,
    ! [I: int,K: nat,J2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J2 )
     => ( ( nth_int @ ( upto @ I @ J2 ) @ K )
        = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).

% nth_upto
thf(fact_10112_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_10113_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_10114_length__upto,axiom,
    ! [I: int,J2: int] :
      ( ( size_size_list_int @ ( upto @ I @ J2 ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J2 @ I ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_10115_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_10116_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_10117_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_10118_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_10119_upto__rec2,axiom,
    ! [I: int,J2: int] :
      ( ( ord_less_eq_int @ I @ J2 )
     => ( ( upto @ I @ J2 )
        = ( append_int @ ( upto @ I @ ( minus_minus_int @ J2 @ one_one_int ) ) @ ( cons_int @ J2 @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_10120_upto__code,axiom,
    ( upto
    = ( ^ [I2: int,J3: int] : ( upto_aux @ I2 @ J3 @ nil_int ) ) ) ).

% upto_code
thf(fact_10121_upto__split3,axiom,
    ! [I: int,J2: int,K: int] :
      ( ( ord_less_eq_int @ I @ J2 )
     => ( ( ord_less_eq_int @ J2 @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J2 @ one_one_int ) ) @ ( cons_int @ J2 @ ( upto @ ( plus_plus_int @ J2 @ one_one_int ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_10122_upto__split2,axiom,
    ! [I: int,J2: int,K: int] :
      ( ( ord_less_eq_int @ I @ J2 )
     => ( ( ord_less_eq_int @ J2 @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ J2 ) @ ( upto @ ( plus_plus_int @ J2 @ one_one_int ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_10123_upto__split1,axiom,
    ! [I: int,J2: int,K: int] :
      ( ( ord_less_eq_int @ I @ J2 )
     => ( ( ord_less_eq_int @ J2 @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J2 @ one_one_int ) ) @ ( upto @ J2 @ K ) ) ) ) ) ).

% upto_split1
thf(fact_10124_distinct__upto,axiom,
    ! [I: int,J2: int] : ( distinct_int @ ( upto @ I @ J2 ) ) ).

% distinct_upto
thf(fact_10125_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ J3 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_10126_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I2: int,J3: int] : ( append_int @ ( upto @ I2 @ J3 ) ) ) ) ).

% upto_aux_def
thf(fact_10127_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_10128_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_10129_upto__rec1,axiom,
    ! [I: int,J2: int] :
      ( ( ord_less_eq_int @ I @ J2 )
     => ( ( upto @ I @ J2 )
        = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J2 ) ) ) ) ).

% upto_rec1
thf(fact_10130_upto_Oelims,axiom,
    ! [X: int,Xa3: int,Y: list_int] :
      ( ( ( upto @ X @ Xa3 )
        = Y )
     => ( ( ( ord_less_eq_int @ X @ Xa3 )
         => ( Y
            = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa3 ) ) ) )
        & ( ~ ( ord_less_eq_int @ X @ Xa3 )
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
thf(fact_10131_upto_Osimps,axiom,
    ( upto
    = ( ^ [I2: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I2 @ J3 ) @ ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_10132_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_10133_upto_Opelims,axiom,
    ! [X: int,Xa3: int,Y: list_int] :
      ( ( ( upto @ X @ Xa3 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa3 ) )
       => ~ ( ( ( ( ord_less_eq_int @ X @ Xa3 )
               => ( Y
                  = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa3 ) ) ) )
              & ( ~ ( ord_less_eq_int @ X @ Xa3 )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa3 ) ) ) ) ) ).

% upto.pelims
thf(fact_10134_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_10135_sort__upto,axiom,
    ! [I: int,J2: int] :
      ( ( linord1735203802627413978nt_int
        @ ^ [X2: int] : X2
        @ ( upto @ I @ J2 ) )
      = ( upto @ I @ J2 ) ) ).

% sort_upto
thf(fact_10136_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_10137_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_10138_integer__of__num__triv_I2_J,axiom,
    ( ( code_integer_of_num @ ( bit0 @ one ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% integer_of_num_triv(2)
thf(fact_10139_sort__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( linord738340561235409698at_nat
        @ ^ [X2: nat] : X2
        @ ( upt @ M @ N ) )
      = ( upt @ M @ N ) ) ).

% sort_upt
thf(fact_10140_tl__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ N ) ) ).

% tl_upt
thf(fact_10141_upt__0__eq__Nil__conv,axiom,
    ! [J2: nat] :
      ( ( ( upt @ zero_zero_nat @ J2 )
        = nil_nat )
      = ( J2 = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
thf(fact_10142_upt__conv__Nil,axiom,
    ! [J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ J2 @ I )
     => ( ( upt @ I @ J2 )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_10143_length__upt,axiom,
    ! [I: nat,J2: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J2 ) )
      = ( minus_minus_nat @ J2 @ I ) ) ).

% length_upt
thf(fact_10144_sorted__list__of__set__range,axiom,
    ! [M: nat,N: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( upt @ M @ N ) ) ).

% sorted_list_of_set_range
thf(fact_10145_upt__eq__Nil__conv,axiom,
    ! [I: nat,J2: nat] :
      ( ( ( upt @ I @ J2 )
        = nil_nat )
      = ( ( J2 = zero_zero_nat )
        | ( ord_less_eq_nat @ J2 @ I ) ) ) ).

% upt_eq_Nil_conv
thf(fact_10146_nth__upt,axiom,
    ! [I: nat,K: nat,J2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J2 )
     => ( ( nth_nat @ ( upt @ I @ J2 ) @ K )
        = ( plus_plus_nat @ I @ K ) ) ) ).

% nth_upt
thf(fact_10147_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_10148_upt__filter__extend,axiom,
    ! [U: nat,U3: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ U @ U3 )
     => ( ! [I3: nat] :
            ( ( ( ord_less_eq_nat @ U @ I3 )
              & ( ord_less_nat @ I3 @ U3 ) )
           => ~ ( P @ I3 ) )
       => ( ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U ) )
          = ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U3 ) ) ) ) ) ).

% upt_filter_extend
thf(fact_10149_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2923211474154528505it_int @ N @ L ) )
     => ( ( map_nat_o @ ( bit_se1146084159140164899it_int @ K ) @ ( upt @ zero_zero_nat @ N ) )
        = ( map_nat_o @ ( bit_se1146084159140164899it_int @ L ) @ ( upt @ zero_zero_nat @ N ) ) ) ) ).

% map_bit_range_eq_if_take_bit_eq
thf(fact_10150_map__decr__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( upt @ M @ N ) ) ).

% map_decr_upt
thf(fact_10151_map__Suc__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_10152_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] :
      ( ( map_nat_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ Ofs )
        @ ( upt @ A @ B ) )
      = ( upt @ ( plus_plus_nat @ A @ Ofs ) @ ( plus_plus_nat @ B @ Ofs ) ) ) ).

% map_add_upt'
thf(fact_10153_map__add__upt,axiom,
    ! [N: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).

% map_add_upt
thf(fact_10154_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_10155_upt__add__eq__append,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( upt @ I @ ( plus_plus_nat @ J2 @ K ) )
        = ( append_nat @ ( upt @ I @ J2 ) @ ( upt @ J2 @ ( plus_plus_nat @ J2 @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_10156_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N4: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ ( suc @ M5 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_10157_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N4 ) ) ) ) ).

% atLeast_upt
thf(fact_10158_distinct__upt,axiom,
    ! [I: nat,J2: nat] : ( distinct_nat @ ( upt @ I @ J2 ) ) ).

% distinct_upt
thf(fact_10159_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N4: nat,M5: nat] : ( set_nat2 @ ( upt @ N4 @ ( suc @ M5 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_10160_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I2: nat,J3: nat] : ( set_nat2 @ ( upt @ I2 @ J3 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_10161_upt__append,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( append_nat @ ( upt @ zero_zero_nat @ I ) @ ( upt @ I @ J2 ) )
        = ( upt @ zero_zero_nat @ J2 ) ) ) ).

% upt_append
thf(fact_10162_upt__conv__Cons,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( upt @ I @ J2 )
        = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J2 ) ) ) ) ).

% upt_conv_Cons
thf(fact_10163_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M @ Q2 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_10164_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N4: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ M5 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_10165_upt__eq__Cons__conv,axiom,
    ! [I: nat,J2: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I @ J2 )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I @ J2 )
        & ( I = X )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J2 )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_10166_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N4 ) ) ) ) ) ).

% atMost_upto
thf(fact_10167_upt__rec,axiom,
    ( upt
    = ( ^ [I2: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J3 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_10168_upt__eq__lel__conv,axiom,
    ! [L: nat,H2: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( ( upt @ L @ H2 )
        = ( append_nat @ Is1 @ ( cons_nat @ I @ Is2 ) ) )
      = ( ( Is1
          = ( upt @ L @ I ) )
        & ( Is2
          = ( upt @ ( suc @ I ) @ H2 ) )
        & ( ord_less_eq_nat @ L @ I )
        & ( ord_less_nat @ I @ H2 ) ) ) ).

% upt_eq_lel_conv
thf(fact_10169_upt__Suc__append,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( upt @ I @ ( suc @ J2 ) )
        = ( append_nat @ ( upt @ I @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_10170_upt__Suc,axiom,
    ! [I: nat,J2: nat] :
      ( ( ( ord_less_eq_nat @ I @ J2 )
       => ( ( upt @ I @ ( suc @ J2 ) )
          = ( append_nat @ ( upt @ I @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I @ J2 )
       => ( ( upt @ I @ ( suc @ J2 ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_10171_int__set__bits__K__False,axiom,
    ( ( bit_bi6516823479961619367ts_int
      @ ^ [Uu3: nat] : $false )
    = zero_zero_int ) ).

% int_set_bits_K_False
thf(fact_10172_take__bit__num__simps_I1_J,axiom,
    ! [M: num] :
      ( ( bit_take_bit_num @ zero_zero_nat @ M )
      = none_num ) ).

% take_bit_num_simps(1)
thf(fact_10173_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_10174_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_10175_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ N @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_10176_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_10177_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_10178_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_10179_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q4 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_10180_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_10181_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_10182_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_10183_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_10184_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one @ one )
    = none_num ) ).

% and_not_num.simps(1)
thf(fact_10185_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_10186_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_10187_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(4)
thf(fact_10188_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(7)
thf(fact_10189_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q2: num] :
      ( ( ( bit_and_not_num @ M @ N )
        = ( some_num @ Q2 ) )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = ( numeral_numeral_int @ Q2 ) ) ) ).

% and_not_num_eq_Some_iff
thf(fact_10190_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N12: num] : ( some_num @ ( bit1 @ N12 ) )
        @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(8)
thf(fact_10191_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( bit_and_not_num @ M @ N )
        = none_num )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
thf(fact_10192_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).

% int_numeral_not_and_num
thf(fact_10193_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).

% int_numeral_and_not_num
thf(fact_10194_bin__last__set__bits,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_bi6516823479961619367ts_int @ F ) ) )
        = ( F @ zero_zero_nat ) ) ) ).

% bin_last_set_bits
thf(fact_10195_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N4: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N4 @ M ) ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_10196_wf__set__bits__int__Suc,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int
        @ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
      = ( bit_wf_set_bits_int @ F ) ) ).

% wf_set_bits_int_Suc
thf(fact_10197_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ N @ one )
      = ( case_nat_option_num @ none_num
        @ ^ [N4: nat] : ( some_num @ one )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_10198_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N4: nat] :
            ( case_o6005452278849405969um_num @ none_num
            @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
            @ ( bit_take_bit_num @ N4 @ M ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_10199_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N4: nat,M5: num] :
          ( produc478579273971653890on_num
          @ ^ [A4: nat,X2: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P3: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
                      @ ( bit_take_bit_num @ O @ P3 ) )
                  @ ^ [P3: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P3 ) ) )
                  @ X2 )
              @ A4 )
          @ ( product_Pair_nat_num @ N4 @ M5 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_10200_int__set__bits__unfold__BIT,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( bit_bi6516823479961619367ts_int @ F )
        = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( F @ zero_zero_nat ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_bi6516823479961619367ts_int @ ( comp_nat_o_nat @ F @ suc ) ) ) ) ) ) ).

% int_set_bits_unfold_BIT
thf(fact_10201_bin__rest__set__bits,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( divide_divide_int @ ( bit_bi6516823479961619367ts_int @ F ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( bit_bi6516823479961619367ts_int @ ( comp_nat_o_nat @ F @ suc ) ) ) ) ).

% bin_rest_set_bits
thf(fact_10202_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_10203_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_10204_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).

% less_eq_nat.simps(2)

% Helper facts (34)
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_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X8: real] : ( P @ X8 ) ) ) ).

thf(help_If_2_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $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__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__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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [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 (8)
thf(conj_0,hypothesis,
    ( tia
    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ x13 @ x14 ) ) ).

thf(conj_1,hypothesis,
    ~ ( ord_less_nat @ ma @ xa ) ).

thf(conj_2,hypothesis,
    ~ ( ord_less_nat @ xa @ mi ) ).

thf(conj_3,hypothesis,
    xa != ma ).

thf(conj_4,hypothesis,
    xa != mi ).

thf(conj_5,hypothesis,
    ord_less_nat @ ( divide_divide_nat @ xa @ ( 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 ) ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).

thf(conj_6,hypothesis,
    ( ( size_s7982070591426661849_VEBTi @ tree_is )
    = ( size_s6755466524823107622T_VEBT @ treeList ) ) ).

thf(conj_7,conjecture,
    entails @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ vEBT_vebt_assn_raw @ treeList @ tree_is ) @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ x13 @ tree_is ) @ ( vEBT_vebt_assn_raw @ summary @ x14 ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ summary @ x14 ) @ ( snga_assn_VEBT_VEBTi @ x13 @ tree_is_103_ATP ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ treeList @ tree_is_103_ATP ) ) ).

%------------------------------------------------------------------------------